Quasiconvex bulk and surface energies with subquadratic growth

1.   Introduction and statements

Let us consider a functional $\mathcal{F}$ with density energy discontinuous through an interface $\partial A$, inside an open bounded subset $\Omega$ of ${\mathbb R}^n$, of the form

where $v\in W^{1, p}_{loc}({\Omega}; {\mathbb R}^N)$, $F, G:{\mathbb R}^{n\times N}\rightarrow {\mathbb R}$ are $C^2$-integrands, $A\subset\Omega$ and $P(A, {\Omega})$ stands for the perimeter of the set $A$ in $\Omega$. Assume that these integrands satisfy the following growth and uniformly strict $p$-quasiconvexity conditions, for $p > 1$ and positive constants $\ell_1, \, \ell_2, \, L_1, \, L_2$:

for every $\xi\in{{\mathbb R}^{n\times N}}$ and $\varphi\in C_0^1(\Omega; \mathbb{R}^N)$.

Existence and regularity results have been obtained initially in the scalar case ($N = 1$) in ^{[4,5,10,17,22,23,24,25,26,29,34,35,36]}. In the vectorial case ($N > 1$), the authors in ^[11] proved the existence of local minimizers of (1.1), for any $p > 1$ under the quasiconvexity assumption quoted above. In the same paper, the $C^{1, \alpha}$ partial regularity is proved for minimal configurations outside a negligible set, in the quadratic case $p = 2$.

In ^[9] the same regularity result has been established in the general case $p\ge 2$, also addressing anisotropic surface energies. Almgren was the first to study such surface energies in his celebrated paper ^[3] (see also ^{[8,21,27,39,40]} for subsequent results). This kind of energies arises in many physical contexts such as the formation of crystals (see ^[6,7]), liquid drops (see ^[16,28]), capillary surfaces (see ^[18,19]) and phase transitions (see ^[33]).

In this paper, we consider the same functional as in ^[9], given by

in the case of sub-quadratic growth, $1 < p < 2$. We achieve analogous regularity results as those established in ^[9], thereby completing the answer to the problem for all $p > 1$.

In this setting $A\subset{\Omega}$ is a set of finite perimeter, $u\in W_{\mathrm{loc}}^{1, p}({\Omega}; \mathbb{R}^N)$, $\mathbb{1}_A$ is the characteristic function of the set $A$, $\partial^* A$ denotes the reduced boundary of $A$ in $\Omega$ and $\nu_A$ is the measure-theoretic outer unit normal to $A$. Moreover, $\Phi$ is an elliptic integrand on $\Omega$ (see Definition 2.8), i.e., $\Phi:\overline{\Omega}\times {\mathbb R}^n\rightarrow [0, \infty]$ is lower semicontinuous, $\Phi(x, \cdot)$ is convex and positively one-homogeneous, $\Phi(x, t\nu) = t\Phi(x, \nu)$ for every $t\geq 0$, and the anisotropic surface energy of a set $A$ of finite perimeter in $\Omega$ is defined as follows

for every Borel set $B\subset\Omega$. The further assumption

with $\Lambda > 1$, allows to compare the surface energy introduced in (1.3) with the usual perimeter. Let us recall that in the vectorial setting, as in the previously cited papers, the regularity we can expect for the gradient of the minimal deformation $u:\Omega\rightarrow {\mathbb R}^N$, ($N > 1$), even in absence of a surface term, is limited to a partial regularity result.

Definition 1.1. We say that a pair $(u, E)$ is a local minimizer of ${\mathcal I}$ in $\Omega$, if for every open set $U\Subset \Omega$ and every pair $(v, A)$, where $v-u\in W_0^{1, p}(U; \mathbb{R}^N)$ and $A$ is a set of finite perimeter with $A\Delta E\Subset U$, we have

Existence and regularity results for local minimizers of integral functionals with uniformly strict $p$-quasiconvex integrand, also in the non autonomous case, have been widely investigated (see ^{[1,2,12,13,14,15,30,31,32,38]}).

Regarding the functional (1.2), the existence of local minimizers is guaranteed by the following theorem, proved in ^[9].

Theorem 1.2. Let $p > 1$ and assume that (F1), (F2), (G1), and (G2) hold. Then, if $v\in W^{1, p}_{\mathrm{loc}}(\Omega; \mathbb{R}^N)$ and $A\subset {\Omega}$ is a set of finite perimeter in ${\Omega}$, for every sequence $\{(v_k, A_k)\}_{k\in{\mathbb N}}$ such that $\{v_k\}$ weakly converges to $v$ in $W^{1, p}_{\mathrm{loc}}(\Omega; \mathbb{R}^N)$ and $\mathbb{1}_{A_k}$ strongly converges to $\mathbb{1}_{{A}}$ in $L^1_{\mathrm{loc}}(\Omega)$, we have

In particular, ${\mathcal I}$ admits a minimal configuration $(u, \mathbb{1}_E)\in W^{1, p}_{\mathrm{loc}}(\Omega; \mathbb{R}^N)\times BV_{\mathrm{loc}}(\Omega; [0, 1])$.

We emphasize that, in particular, the previous theorem implies the semicontinuity of the anisotropic perimeter functional (1.3) (see ^[9] Proposition 3.2 for the proof).

In this paper, we obtain a $C^{1, \alpha}$ regularity result for local minimizers of (1.2) in the case of sub-quadratic growth, $1 < p < 2$. If we further assume a closeness condition on $F$ and $G$ (see assumption (H) in Theorem 1.3), we prove that $u\in C^{1, \gamma}({\Omega}_1)$ for every $\gamma\in (0, \frac{1}{p'})$ on a full measure set ${\Omega}_1\subset {\Omega}$. Furthermore, we do not assume any regularity on $\Phi$ in order to get the regularity of $u$.

Our main theorem is the following:

Theorem 1.3. Let $(u, E)$ be a local minimizer of ${\mathcal I}$. Let the bulk density energies $F$ and $G$ satisfy (F1), (F2), (G1), and (G2), with $1 < p < 2$, and let the surface energy ${\mathbf{\Phi}}$ be of general type (1.3) with $\Phi$ satisfying (1.4). Assume in addition that

then there exists an open set $\Omega_1\subset \Omega$ of full measure such that $u\in C^{1, \gamma}({\Omega}_1;\mathbb{R}^N)$ for every $\gamma\in \big(0, \frac{1}{p'}\big)$.

In the case where hypothesis $(H)$ does not hold, it is still possible to establish a partial $C^{1, \beta}$ regularity result. To avoid redundancy and overlap, we have chosen to present this result in the form of a remark. Nevertheless, throughout the paper, we will provide some sketches and insights into the proof in this case as well.

Remark 1.4. We remark that if $(u, E)$ is a local minimizer of ${\mathcal I}$ with the bulk density energies $F$ and $G$ satisfying (F1), (F2), (G1), (G2), $1 < p < 2$, and the surface energy ${\mathbf{\Phi}}$ of general type (1.3) satisfying (1.4), then there exist an exponent $\beta\in (0, 1)$ and an open set $\Omega_0\subset \Omega$ with full measure such that $u\in C^{1, \beta}({\Omega}_0;\mathbb{R}^N)$.

The proof of the Theorem 1.3 is based on a blow-up argument aimed to establish a decay estimate for the excess function

where

To this aim, we use a comparison argument between the blow-up sequence $v_h$ at small scale in the balls $B_{r_h}(x_h)$ and the solution $v$ of a suitable linearized system. The challenging part of the argument, as usual, is to prove that the 'good' decay estimates available for the function $v$ (see Proposition 2.1), are inherited by the $v_h$ as $h\rightarrow \infty$.

To achieve this result, the main tool is a Caccioppoli type inequality that we prove for minimizers of perturbed rescaled functionals (see (3.16)) involving the function $V(Dv_h)$ and the perimeter of the rescaled minimal set $E_h$. The Caccioppoli inequality combined with the Sobolev-Poincaré inequality will lead us to a contradiction (see Step 6 of Proposition 3.1). In this final step, the issue to deal with the function $V(Du)$ in the sub-quadratic case, is overcome by using a suitable Sobolev Poincaré inequality involving $V(Du)$ (see Theorem 2.6), whose proof is due to ^[12].

2.   Preliminaries

Let $\Omega$ be a bounded open set in ${\mathbb R}^n$, $n\geq 2$, $u:\Omega\rightarrow {\mathbb R}^N$, $N > 1$. We denote by $B_r(x): = \left\{y\in {\mathbb R}^n: |y-x| < r\right\}$ the open ball centered at $x\in {\mathbb R}^n$ of radius $r > 0$, $\mathbb S^{n-1}$ represents the unit sphere of ${\mathbb R}^n$, $c$ a generic constant that may vary.

For $B_r(x_0)\subset{\mathbb R}^n$ and $u\in L^1(B_r(x_0);{\mathbb R}^N)$ we denote

and we will omit the dependence on the center when it is clear from the context. We denote by $|\cdot|$ the standard Euclidean norm, defined as

for every $\xi\in \mathbb{R}^{n\times N}.$

If $F \colon {{\mathbb R}^{n\times N}} \to {\mathbb R}$ is sufficiently differentiable, we write

for $\xi$, $\eta \in {{\mathbb R}^{n\times N}}$.

It is well known that for quasiconvex $C^1$ integrands the assumptions (F1) and (G1) yield the upper bounds

for all $\xi \in {{\mathbb R}^{n\times N}}$, with $c_1$ and $c_2$ constants depending only on $p$ (see [32, Lemma 5.2] or ^[38]).

Furthermore, if $F$ and $G$ are $C^2$, then (F2) and (G2) imply the following strong Legendre-Hadamard conditions

for all $Q\in{{\mathbb R}^{n\times N}}$, $\lambda\in {\mathbb R}^n$, $\mu \in {\mathbb R}^N$, where $c_3 = c_3(p, \ell_1)$ and $c_4 = c_4(p, \ell_2)$ are positive constants (see [32, Proposition 5.2]). Throughout the paper, we frequently employ the Einstein summation convention. We will need the following quite standard regularity result (see ^[12] for its proof).

Proposition 2.1. Let $v\in W^{1, 1}(\Omega; {\mathbb R}^N)$ be such that

for every $\varphi \in C_{c}^\infty(\Omega; {\mathbb R}^N)$, where $Q = \{Q^{ij}_{\alpha\beta}\}$ is a constant matrix satisfying $|Q^{ij}_{\alpha \beta}|\leq L$ and the strong Legendre-Hadamard condition

for all $\lambda\in{\mathbb R}^n$, $\mu\in{\mathbb R}^N$ and for some positive constants $\ell, L > 0$. Then $v \in C^\infty$ and, for any $B_ R (x_0) \subset \Omega$, the following estimate holds

where $c = c(n, N, \ell, L) > 0$.

We assume that $1 < p < 2$ and we refer to the auxiliary function

whose useful properties are listed in the following lemma (see ^[12] for the proof).

Lemma 2.2. Let $1 < p < 2$ and let $V:{\mathbb R}^k\rightarrow {\mathbb R}^k$ be the function defined in (2.2), then for any $\xi, \eta \in {\mathbb R}^k$ and $t > 0$ the following inequalities hold:

(i) $2^{(p-2)/4}\min\{|\xi|, |\xi|^{p/2}\}\leq \bigl|V(\xi)\bigr|\leq \min\{|\xi|, |\xi|^{p/2}\}$,

(ii) $\bigl|V(t\xi)\bigr|\leq \max\{t, t^{p/2}\}\bigl|V(\xi)\bigr|$,

(iii) $\bigl|V(\xi+\eta)\bigr|\leq c(p)\bigl[\bigl|V(\xi)\bigr|+\bigl|V(\eta)\bigr|\bigr]$,

(iv) $\frac p2|\xi-\eta|\leq \bigl(1+|\xi|^2+|\eta|^2\bigr)^{(2-p)/4}\bigl|V(\xi)-V(\eta)\bigr|\leq c(k, p)|\xi-\eta|$,

(v) $\bigl|V(\xi)-V(\eta)\bigr|\leq c(k, p)\bigl|V(\xi-\eta)\bigr|$,

(vi) $\bigl|V(\xi-\eta)\bigr|\leq c(p, M)\bigl|V(\xi)-V(\eta)\bigr|$, if $|\eta| \leq M$.

We will also use the following iteration lemma (see [32, Lemma 6.1]).

Lemma 2.3. Let $0 < \rho < R$ and let $\psi \colon [\rho, R] \to {\mathbb R}$ be a bounded non negative function. Assume that for all $\rho \leq s < t \leq R$ we have

where $\vartheta \in [0, 1)$, $\alpha > \beta > 0$ and $A, B, C \geq 0$ are constants. Then there exists a constant $c = c(\vartheta, \alpha) > 0$ such that

An easy extension of this result can be obtained by replacing homogeneity with condition (ⅱ) of Lemma 2.2.

Lemma 2.4. Let $R > 0$ and let $\psi \colon [R/2, R] \to [0, +\infty)$ be a bounded function. Assume that for all $R/2 \leq s < t \leq R$ we have

where $h\in L^p(B_r)$, $A, B > 0$, and $0 < \vartheta < 1$. Then there exists a constant $c(\vartheta) > 0$ such that

Given a $C^1$ function $f:{\mathbb R}^k\to {\mathbb R}$, $Q\in {\mathbb R}^k$ and $\lambda > 0$, we set

In the next sections we will use the following lemma about the growth of $f_{Q, \lambda}$ and $Df_{Q, \lambda}$.

Lemma 2.5. Let $1 < p < \infty$, and let $f$ be a $C^{2}({\mathbb R}^k)$ function such that

for any $\xi\in{\mathbb R}^k$ and for some $L > 0$. Then for every $M > 0$ there exists a constant $c = c(p, L, M) > 0$ such that, for every $Q\in {\mathbb R}^k$, $|Q|\leq M$ and $\lambda > 0$, it holds

for all $\xi\in{\mathbb R}^k$.

Proof. Applying Taylor's formula for every $\xi \in {\mathbb R}^k$, there exists $\theta\in [0, 1]$ such that,

If we denote $K_M: = \max\left\{|D^2 f(\xi)|: |\xi|\leq M+1\right\}$, we have

On the other hand, using growth condition (2.3) and the definitions of $f_{Q, \lambda}$ and $Df_{Q, \lambda}$, we get

We get the result by combining (2.5) and (2.6). □

A fundamental tool in order to handle the subquadratic case is the following Sobolev-Poincaré inequality related to the function $V$, as established in Theorem $2.4$ of ^[12].

Theorem 2.6. If $1 < p < 2$, there exist $2/p < \alpha < 2$ and $\sigma > 0$ such that if $u \in W^{1, p}(B_{3R}(x_0), {\mathbb R}^N)$, then

where the positive constant $C = C(n, N, p)$ is independent of $R$ and $u$.

We remark that a sharper version of Theorem 2.6 can be found in ^[20].

In the remaining part of this section, we recall some elementary definitions and well-known properties of sets of finite perimeter. We introduce the notion of anisotropic perimeter as well.

Given a set $E\subset {\mathbb R}^n$ and $t\in [0, 1]$, we define the set of points of $E$ of density $t$ as

Let $U$ be an open subset $U$ of ${\mathbb R}^n$. A Lebesgue measurable set $E\subset {\mathbb R}^n$ is said to be a set of locally finite perimeter in $U$ if there exists a ${\mathbb R}^n$-valued Radon measure $\mu_E$ on $U$ (called the Gauss-Green measure of $E$) such that

Moreover, we denote the perimeter of $E$ relative to $G\subset U$ by $P(E, G) = |\mu_E|(G)$.

It is well known that the support of $\mu_E$ can be characterized by

(see [37, Proposition 12.19]). If $E$ is of finite perimeter in $U$, the reduced boundary $\partial^*E \subset U$ of $E$ is the set of those $x\in U$ such that

exists and belongs to $\mathbb S^{n-1}$. The essential boundary of $E$ is defined as $\partial^e E: = {\mathbb R}^n\setminus(E^{(0)}\cup E^{(1)})$. It is well-understood that

Furthermore, Federer's criterion (see for instance [37, Theorem 16.2]) ensures that

By De Giorgi's rectifiability theorem (see [37, Theorem 15.9]), the Gauss-Green measure $\mu_E$ is completely characterized as follows:

The equality holds in the class of Borel sets compactly contained in $U$. Here, we have denoted $\mu\llcorner \partial^*E(F) = \mu (\partial^*E \cap F)$, for any subset $F$ of ${\mathbb R}^n.$

Remark 2.7 (Minimal topological boundary). If $E\subset{\mathbb R}^n$ is a set of locally finite perimeter in $U$ and $F\subset{\mathbb R}^n$ is such that $|(E\Delta F)\cap U| = 0$, then $F$ is a set of locally finite perimeter in $U$ and $\mu_E = \mu_F$. In the rest of the paper, the topological boundary $\partial E$ must be understood by considering the suitable representative of $E$ in order to have that $\overline{\partial^*E} = \partial E\cap U$. We will choose $E^{(1)}$ as representative of $E$. With such a choice it can be easily verified that

Therefore, by (2.8),

In what follows, we give the definition of anisotropic surface energies and we recall some properties.

Definition 2.8 (Elliptic integrands). Given an open subset $\Omega$ of ${\mathbb R}^n$, $\Phi:\overline{\Omega}\times {\mathbb R}^n\rightarrow [0, \infty]$ is said to be an elliptic integrand on $\Omega$ if it is lower semicontinuous, with $\Phi(x, \cdot)$ convex and positively one-homogeneous for any $x\in\overline{\Omega}$, i.e., $\Phi(x, t\nu) = t\Phi(x, \nu)$ for every $t\geq 0$. Accordingly, the anisotropic surface energy of a set $E$ of finite perimeter in $\Omega$ is defined as

for every Borel set $B\subset\Omega$.

In order to prove the regularity of minimizers of anisotropic surface energies, it is well known that a $C^k$-dependence of the integrand $\Phi$ on the variable $\nu$, and a continuity condition with respect to the variable $x$, must be assumed (see the seminal paper ^[3]). In fact, one more condition is essential, that is a non-degeneracy type condition for the integrand $\Phi$. More precisely, we have to assume that there exists a constant $\Lambda > 1$ such that

for any $x\in\Omega$ and $\nu\in\mathbb{S}^{n-1}$. We emphasize that (2.10) is the only assumption we make for the elliptic integrand $\Phi$. We observe that, if the elliptic integrand $\Phi$ satisfies the previous condition, then the anisotropic surface energy (2.9) satisfies the following comparability condition to the perimeter:

for any set $E$ of finite perimeter in $\Omega$ and any Borel set $B\subset\Omega$.

A useful relation is given by proposition below proved in ^[9].

Proposition 2.9. Let $U\subset{\mathbb R}^n$ be an open set and let $E, F\subset U$ be two sets of finite perimeter in $U$. It holds that

3.   Decay estimates

In this section we prove decay estimates for local minimizers $u$ of the functionals (1.2), see Definition 1.1, by using a well-known blow-up technique involving a suitable excess function. We consider the bulk excess function defined as

for $B_r(x_0)\subset\Omega$.

When the assumption $(H)$ is in force, we refer to the following "hybrid" excess:

Proposition 3.1. Let $(u, E)$ be a local minimizer of the functional $\mathcal{I}$ in (1.2) and let the assumptions (F1), (F2), (G1), (G2), and (H) hold. For every $M > 0$ and every $0 < \tau < \frac{1}{4}$, there exist two constants $\varepsilon_0 = \varepsilon_0(\tau, M) > 0$ and $C_* = C_*(n, p, \ell_1, \ell_2, L_1, L_2, \Lambda, M) > 0$ such that if for some ball $B_r(x_0)\Subset{\Omega}$ the following condition hold: $|(Du)_{x_0, r}|\leq M\; \mathrm{and}\; U_*(x_{0}, r)\leq \varepsilon_0,$ then

Proof. In order to prove (3.2), we argue by contradiction. Let $M > 0$ and $\tau\in (0, 1/4)$ be such that for every $h\in \mathbb{N}$, $C_* > 0$, there exists a ball $B_{ r_h}(x_h)\Subset \Omega$ such that

and

The constant $C_*$ will be determined later. We remark that we can confine ourselves to the case in which $E\cap B_{ r_h}(x_h)\not = \emptyset$, since the case in which $B_{ r_h}(x_h)\subset \Omega\setminus E$ is well known, being $U_* = U+r$.

Step 1. Blow-up. We set $\lambda_h^2: = U_*(x_h, r_h)$, $A_h: = (Du)_{x_h, r_h}$, $a_h: = (u)_{x_h, r_h}$, and we define

One can easily check that $(Dv_h)_{0, 1} = 0$ and $(v_h)_{0, 1} = 0$. We set

By using $(ii)$ and $(vi)$ of Lemma 2.2, we deduce

Then, since the integral in the last expression appear in the definition of the excess $U_*(x_h, r_h)$,

it follows that $r_h\to 0$, $P(E_h, B_1)\to 0,$ and

Therefore, by (3.3) and (3.6), using also (ⅰ) of Lemma 2.2 and Poincaré inequality, we deduce that there exist a (not relabeled) subsequence of $\{v_h\}_{h\in{\mathbb N}}$, $A\in {{\mathbb R}^{n\times N}}$ and $v\in W^{1, p}(B_1;{{\mathbb R}^{N}})$, such that

where we have used the fact that $(v_h)_{0, 1} = 0$. Moreover, by (3.3) and (3.6), we have that for every $0\leq\epsilon < \frac{1}{n-1}$

where we have used (3.6) and the choice of $\epsilon < \frac{1}{n-1}$ in the last inequalities. Therefore, by the relative isoperimetric inequality,

In the sequel the proof will proceed differently depending on

The first case is easier to handle. To understand the reason, let us introduce the expansions of $F$ and $G$ around $A_h$ as follows:

for any $\xi\in{\mathbb R}^{n\times N}$. In the first case the suitable rescaled functional to consider in the blow-up procedure is the following:

We claim that $v_h$ satisfies the minimality inequality

for any $\psi\in W^{1, p}_0(B_1;{\mathbb R}^N)$. Indeed, using the minimality of $(u, E)$ with respect to $(u+\varphi, E)$, for $\varphi\in W^{1, p}_0(B_{r_h}(x_h); {\mathbb R}^N)$, where $\varphi$ is defined by the change of variable $y = \frac{x-x_h}{r_h}$, setting $\varphi(x): = \lambda_hr_h\psi(\frac{x-x_h}{r_h})$, it holds that

and (3.12) follows by the definition of ${\cal I}_h$ in (3.11).

In the second case, the suitable rescaled functional to consider in the blow-up procedure is

We claim that

for all $\psi\in W^{1, p}_0(B_1;{\mathbb R}^{N})$. Indeed, the minimality of $(u, E)$ with respect to $(u+\varphi, E)$, for $\varphi\in W^{1, p}_0(B_{r_h}(x_h); {\mathbb R}^{N})$, implies that

where we used that the last integral vanishes outside the support of $\varphi$ and that $G\ge 0$. Using the change of variable $x = x_h+ r_hy$ in the previous formula, we get

or, equivalently, using the definitions of $v_h$,

where ${\psi(y)}: = \frac{\varphi(x_h+r_hy)}{\lambda_h r_h}$, for $y\in B_1$. Therefore, setting

by the definitions of $F_h$ and $G_h$ in (3.10) and using the assumption (G1), we have that

i.e., (3.13).

Step 2. A Caccioppoli type inequality. The key ingredient in our proof is the following Caccioppoli-type inequality. The version presented here, which involves the auxiliary function $V$, was used in ^[12] to address the subquadratic case $1 < p < 2$. In our setting, there is also a perimeter term, which is a distinctive feature of our problem. We also draw attention to ^[20], where a suitable variant of the Caccioppoli-type inequality involving a modified auxiliary function $V_{|A|}$ was established to handle potential degeneracy of the strict quasiconvexity.

We claim that there exists a constant $c = c(n, p, \ell_1, \ell_2, L_1, L_2, M) > 0$ such that for every $0 < \rho < 1$ there exists $h_0 = h_0(n, p, M, \rho)\in \mathbb{N}$ such that

for all $h > h_0$. We divide the proof into two steps.

Substep 2.a The case $\min\{|E^*_h|, |B_1\setminus E_h|\} = |E^*_h|$. We consider $0 < \frac{\rho}{2} < s < t < \rho < 1$ and let $\eta\in C^\infty_0(B_{t})$ be a cut off function between $B_s$ and $B_t$, i.e., $0\leq \eta\leq 1$, $\eta\equiv 1$ on $B_s$ and $|\nabla \eta |\leq \frac{c}{t-s}$. Set $p_h : = (v_h)_{B_\rho}$, $P_h : = (Dv_h)_{B_{\frac{\rho}{2}}}$, and set

for any $y\in B_1$. Proceeding similarly as in (3.10), we rescale $F$ and $G$ around $A_h+\lambda_h P_h$,

for any $\xi\in {\mathbb R}^{n\times N}$. By Lemma 2.5, two growth estimates on $\widetilde F_h$, $\widetilde G_h$ and their gradients hold with some constants that depend on $p, L_1, L_2, M$ (see (3.3)) and could also depend on $\rho$ through $|\lambda_h P_h |$. However, given $\rho$, we may choose $h_0 = h_0(n, p, M, \rho)$ large enough to have

for any $h\geq h_0$. Indeed, by (3.6) the sequence $\left\{D v_h\right\}_h$ is equibounded in $L^p(B_1)$, then we have

and so the constant in (2.4) can be taken independently of $\rho$.

Set

By the uniformly strict quasiconvexity of $\widetilde F_h$, we have

We estimate separately the two addends in the right-hand side of the previous chain of inequalities. We deal with the first addend by means of a rescaling of the minimality condition of $(u, E)$. Using the change of variable $x = x_h+ r_hy$, the fact that $G\ge 0$ and the minimality of $(u, E)$ with respect to $(u+\varphi, E)$ for $\varphi\in W^{1, p}_0(B_{r_h}(x_h); {\mathbb R}^{N})$, we have

i.e., by the definitions (3.5) and (3.17) of $v_h$ and $w_h$, respectively,

for $\psi: = \frac{\varphi(x_h+r_hy)}{\lambda_h r_h}\in W^{1, p}_0(B_1;{\mathbb R}^{N})$. Therefore, recalling the definitions of $\widetilde F_h$ and $\widetilde G_h$ in (3.18), we have that

Choosing $\varphi \; \mathrm{such\; that}\; \psi = -\psi_{1, h}$, the previous inequality becomes

where we have used Lemma 2.5, the second estimate in (2.1), and the fact that $|A_h+\lambda_h P_h|\leq M+1$. By applying Hölder's and Young's inequalities, we get

The previous chain of inequalities combined with (3.20) yields

Now we estimate the second addend in the right-hand side of (3.19). Using the upper bound on $D\widetilde F_h$ in Lemma 2.5,

Regarding the integrand in the latest estimate, we distinguish two cases:

Case 1: $|D\psi_{2, h}|\leq|Dw_h-\theta D\psi_{2, h}|$. By the definition of $V$, we have

Case 2: $|Dw_h-\theta D\psi_{2, h}| < |D\psi_{2, h}|$. If $|D\psi_{2, h}| < 1/\lambda_h$, using (i) of Lemma 2.2 we get

If $|D\psi_{2, h}|\geq 1/\lambda_h$, using again (ⅰ) of Lemma 2.2, we deduce that

By combining the two previous cases, we can proceed in the estimate (3.22) as follows:

Hence, combining (3.19) with (3.21) and (3.23), we obtain

By the definition of $\psi_{2, h}$ and (ii) and (iii) of Lemma 2.2, we infer that

for some $\tilde{C} = \tilde{C}(n, p, L_1, L_2, M)$

By adding $\tilde{C}\int_{B_s}|V(\lambda_h D w_h)|^2\, dy$ to both sides of the previous estimate, dividing by $\ell_1+\tilde{C}$ and thanks to Lemma 2.4, we deduce that

Therefore, by the definition of $w_h$, we conclude that

which, by the relative isoperimetric inequality and the hypothesis of this substep, i.e.,

yields the estimate (3.16).

Substep 2.b The case $\min\{|E^*_h|, |B_1\setminus E_h|\} = |B_1\setminus E_h|$.

Let us fix $0 < \frac{\rho}{2} < s < t < \rho < 1$ and let $\eta\in C^\infty_0(B_{t})$, $p_h$, $P_h$ as in Substep 2.a and define

and

We remark that Lemma 2.5 can be applied to $\widetilde H_h$, that is

and, by the uniformly strict quasiconvexity conditions (F2) and (G2),

where we have set

We set again

By the quasiconvexity condition (3.24) and since $\widetilde H_h(0) = 0$, we have

Similarly to the previous case, we estimate separately the two addends in the right-hand side of the previous chain of inequalities. Using the minimality condition (3.15) for the rescaled functions $v_h$ and recalling the definition of $\tilde{H}_h$, since $Dv_h = Dw_h+P_h$, we get

Choosing $\psi = -\psi_{1, h}$ as test function in (3.26) and using the fact that $\widetilde H_h(0) = 0$, we estimate

We note that, since $|A_h+\lambda_hP_h|\leq c(M)$, for every fixed $\varepsilon > 0$ there exists a constant $C = C(p, \varepsilon)$ such that

Summarizing, we get

Now we estimate the second addend in the right-hand side of (3.25). Using the upper bound on $D\widetilde H_h$ in Lemma 2.5, we obtain

Proceeding exactly as in the estimate (3.23) of the step $2.a$, we obtain

Inserting (3.27) and (3.28) in (3.25), we infer that

Taking advantage of the hole filling technique as in the previous case, we obtain

The assumption (H) implies that there exists $\varepsilon = \varepsilon(p, \ell_1, \ell_2, L_2) > 0$ such that $\frac{(1+\varepsilon)L_2}{\ell_1+\ell_2} < 1$. Therefore we have

So, by virtue of Lemma 2.4, from the previous estimate we deduce that

By definition of $w_h$ and the relative isoperimetric inequality, since $|B_1\setminus E_h| = \min\{|E^*_h|, |B_1\setminus E_h|\}$, we get the estimate (3.16).

Step 3. $v$ solves a linear system in $B_1$.

Let us divide the proof into two cases, depending on which one is the smallest between $|E^*_h|$ and $|B_1\setminus E_h|$.

We divide the proof in two substeps.

Substep 3.a The case $\min\{|E^*_h|, |B_1\setminus E_h|\} = |E^*_h|$.

We claim that $v$ solves the linear system

for all $\psi \in C^1_0(B_1;{\mathbb R}^N)$. Since $v_h$ satisfies (3.12), we have that

for every $\psi \in C^{1}_{0}(B_{1};{\mathbb R}^N)$ and $s\in (0, 1)$. Dividing by $s$ and passing to the limit as $s\to 0$, by the definition of ${\cal I}_{h}$, we get (see ^[9] or [11, Substep 3.a])

We partition the unit ball as follows:

By (3.6), we get

We rewrite (3.29) as follows:

By growth condition in (2.1) and Hölder's inequality, we get

thanks to (3.3), (3.6) and (3.30). Thus

By (3.3) and the definition of $\mathbf{B}_h^-$ we have that $|A_h+\lambda_hDv_h|\leq M+1$ on $\mathbf{B}_h^-$. Hence we estimate

where we have used (3.6). Since, by (3.7), $\lambda_hDv_h\to 0$ a.e. in $B_{1}$, the uniform continuity of $D^2F$ on bounded sets and the Severini-Egorov's Theorem implies that

Note that (3.30) yields that $\mathbb{1}_{\mathbf{B}_h^-}\to \mathbb{1}_{B_1}$ in $L^r(B_1)$, for every $r < \infty$. Therefore, by the weak convergence of $D v_h$ to $D v$ in $L^p(B_1)$, it follows that

By growth condition (2.1), we deduce

where we have used (3.3) and (3.6). Since $\min\{|E^*_h|, |B_1\setminus E_h|\} = |E^*_h|$, by (3.9), we have

and so

By (3.32)–(3.35), passing to the limit as $h\to \infty$ in (3.31), we get

Furthermore, plugging $-\psi$ in place of $\psi$, we get

i.e., $v$ solves a linear system with constant coefficients.

Substep 3.b The case $\min\{|E^*_h|, |B_1\setminus E_h|\} = |B_1\setminus E_h|$.

We claim that $v$ solves the linear system

for all $\psi\in C^1_0(B_1;{\mathbb R}^N)$.

Arguing as in (3.14) and rescaling, we have that

for every $\psi \in C^{1}_{0}(B_{1};{\mathbb R}^N)$ and for every $s\in (0, 1)$. Therefore

Dividing by $s$ and passing to the limit as $s\to 0$, by the definition of ${\cal H}_{h}$ we get

As before, we partition $B_1$ as follows:

We rewrite (3.36) as

Arguing as in (3.32), we obtain that

and, as in (3.33) and (3.34),

Moreover, we have that

where we used (3.6). Since $\min\{|E^*_h|, |B_1\setminus E_h|\} = |B_1\setminus E_h|$, by (3.12), we have

and we obtain

By (3.38) and (3.39), passing to the limit as $h\to\infty$ in (3.37) we conclude that

and, with $-\psi$ in place of $\psi$, we finally get

as claimed.

Substep 3.c. A decay estimate for $Dv$.

By Proposition 2.1 and the theory of linear systems (see [30, Theorem 2.1 and Chapter 3]), we deduce in both cases that $v \in C^\infty$ and there exists a constant $\tilde{c} = \tilde{c}(n, N, p, \ell_1, \ell_2, L_1, L_2) > 0$ such that

for any $\tau\in\big(0, \frac{1}{2}\big)$. Moreover, by Proposition 2.1 again,

Observing that

it follows that

for some fixed $\overline{C} = \overline{C}(n, N, p, \ell_1, \ell_2, L_1, L_2)$.

Step 4. An estimate for the perimeters.

Our aim is to show that there exists a constant $c = c(n, p, L_2, \Lambda, M) > 0$ such that

By the minimality of $(u, E)$ with respect to $(u, \widetilde E)$, where $\widetilde E$ is a set of finite perimeter such that $\widetilde E\Delta E\Subset B_{r_h}(x_h)$ and $B_{r_h}(x_h)$ are the balls of the contradiction argument, we get

Using the change of variable $x = x_h+r_h y$ and dividing by $r_h^{n-1}$, we have

where

for every Borel set $V\subset\Omega$. Assume first that $\min\{ |B_1\setminus E_h|, |E^*_h|\} = |B_1\setminus E_h|$. Choosing $\widetilde E_h: = E_h\cup B_\rho$, we get

By the coarea formula, the relative isoperimetric inequality, the choice of the representative $E^{(1)}_h$ of $E_h$, which is a Borel set, we get

Therefore, thanks to Chebyshev's inequality, we may choose $\rho\in (\tau, 2\tau)$, independent of $h$, such that, up to subsequences, it holds

We remark that Proposition 2.9 holds also for ${\mathbf{\Phi}}_h$. Thus, thanks to the choice of $\rho$, being $\mathcal{H}^{n-1}(\partial^*E_h\cap \partial B_{\rho}) = 0$, we have that

By the choice of the representative of $E_h$ (see Remark 2.7), taking into account (2.10) and the inequality in (3.43), it follows that

On the other hand, by (2.10) and the additivity of the measure ${\mathbf{\Phi}}_h(E_h, \cdot)$ it holds that

since $\rho > \tau$. Combining (3.42), (3.44) and (3.45), we obtain

where we used (3.6). The previous estimate leads to (3.41). We reach the same conclusion if

choosing $\widetilde E_h = E_h\setminus B_\rho$ as a competitor set.

Step 5. Higher integrability of $v_h$.

We will prove that there exist two positive constants $C$ and $\delta$ depending on $n, p, \ell_1, \ell_2, L_1, L_2$ such that for every $B_r\subset B_1$ it holds

We remark that, using (2.4) in Lemma 2.5 and (ⅳ) of Lemma 2.2,

and

Let $r > 0$ be such that $B_{3r}\subset B_1$, $\frac{r}{2} < s < t < r$ and $\eta\in C^1_c(B_t)$ be such that $0\leq\eta\leq 1$, $\eta = 1$ on $B_s$, $|D\eta|\leq \frac{c}{t-s}$, for some positive constant $c$. We define

We deal with the case $\min\{|E^*_h|, |B_1\setminus E_h|\} = |E^*_h|$, the other one is similar. Using the fact that $G_h\geq 0$ and the minimality relation (3.12) we deduce

Then, using growth condition (3.48) and the fact that $A_h$ is controlled by $M$, we conclude that

By the properties of $V$, it holds that

Thus, using Young's inequality, it follows that

for some $\varepsilon > 0$ to be chosen. Combining the previous two chains of inequalities, we deduce that

Choosing $\varepsilon$ sufficiently small, we absorb the last integral to the left-hand side

By (ii) and (iii) of Lemma 2.2, it follows

By applying the hole-filling technique, we add $c(p, \ell_1, L_1, L_2, M)\int_{B_s}|V(\lambda_h D v_h)|^2\, dy$, and we get

Now we can apply Lemma 2.4 and derive

Finally, by Hölder's inequality and Theorem 2.7 we gain

We conclude the proof by applying Gehring's lemma (see [32, Theorem 6.6]).

Step 6. Conclusion.

By the change of variable $x = x_h+r_h y$, inequalities (3.6), (3.7) and (v) of Lemma 2.2, for every $0 < \tau < \frac{1}{4}$, we have

Then, using Caccioppoli inequality in (3.16) and estimate of the perimeter (3.46), we get

where we have used (3.6), (3.8) and estimate (3.46).

Now we want to prove the following extimate:

The last inequality is obtained by using that $v$ and $Dv$ are bounded, $\lambda_ h\rightarrow 0$ and $|V(\xi)|\leq|\xi|$ for $|\xi|\leq 1$.

We observe that proving the equality is equivalent to show

In the sequel $\sigma$ will denote the exponent given in the Sobolev-Poincaré type inequality of the Theorem 2.7. We can assume that the higher integrability exponent $\delta$ given in the Step 5 is such that $\delta < \sigma$.

Let us choose $\theta\in(0, 1)$ such that $2\theta+\frac{1-\theta}{1+\sigma} = 1$. Applying Hölder's inequality, it holds that

Using the fact that $|V(\xi)|\leq |\xi|$ and (iii) of Lemma 2.2, for the first factor in the previous product, and using also Theorem 2.7 applied to $(v_h-v) -(v_h-v)_{2\tau}-(Dv_h-Dv)_{\tau}\, y$, we deduce

with $2/p < \alpha < 2$ given in Theorem 2.7.

In the last term we can increase choosing $\alpha = 2$, moreover, using again (iii) of Lemma 2.2 we deduce

In the last term, we observe that the second addend can be estimated by making use of (i) of Lemma 2.2, the fact that $Dv_h \rightharpoonup Dv$ weakly in $L^p(B_1, \mathbb{R}^{nN})$ and $\lambda_h \to 0$. In particular, we obtain

Regarding the term

using (3.47) and the definition of $v_h$, we deduce

where $0\leq\epsilon < \frac{1}{n-1}$. Therefore, by Hölder's inequality, we have

We conclude that

By virtue of (3.6), (3.8), (3.9), the Poincaré-Wirtinger inequality and (3.40), we get

The contradiction follows, by choosing $C_*$ such that $C_* > C$, since, by (3.5),

□

If assumption (H) is not taken into account, it is still possible to establish a decay result for the excess, analogous to the previous one. However, this requires employing a modified ''hybrid" excess, defined as:

where $U(x_0, r)$ is defined in (3.1), $\delta$ is the higher integrability exponent given in the Step 5 of Proposition 3.1 and $0 < \beta < \frac{\delta}{1+\delta}$. The following result still holds true.

Proposition 3.2. Let $(u, E)$ be a local minimizer of $\mathcal{I}$ in (1.2) under the assumptions (F1), (F2), (G1), and (G2). For every $M > 0$ and $0 < \tau < \frac{1}{4}$, there exist two positive constants $\varepsilon_0 = \varepsilon_0(\tau, M)$ and $c_{**} = c_{**}(n, p, \ell_1, \ell_2, L_1, L_2, \Lambda, \delta, M)$ for which, whenever $B_r(x_0)\Subset{\Omega}$ verifies

then

In order to avoid unnecessary repetition we do not include the proof here, as it is almost identical to the proof of the Proposition 3.1, with the obvious adjustments, see ^[9].

4.   Proof of the main theorem

Here we give the proof of Theorem 1.3 through a suitable iteration procedure. It is easy to show the validity of the following lemma by arguing exactly in the same way as in [11, Lemma 6.1].

Lemma 4.1. Let $(u, E)$ be a local minimizer of the functional $\mathcal{I}$ and let $c_*$ the constant introduced in Proposition 3.1. For every $\alpha\in (0, 1)$ and $M > 0$ there exists $\vartheta_0 = \vartheta_0(c_*, \alpha) < 1$ such that for $\vartheta\in (0, \vartheta_0)$ there exists a positive constant $\varepsilon_1 = \varepsilon_1(n, p, \ell_1, \ell_2, L_1, L_2, M, \vartheta)$ such that, if $B_r(x_0)\Subset \Omega$,

then

Proof. Let $M > 0$, $\alpha\in (0, 1)$ and $\vartheta\in (0, \vartheta_0)$, where $\vartheta_0 < 1$. Let $\varepsilon_1 < \varepsilon_0$, where $\varepsilon_0$ is the constant appearing in Proposition 3.1. We first prove by induction that

If $h = 0$, the statement holds. Assuming that (4.1) holds for $h > 0$, applying properties (i) and (vi) of Lemma 2.2, we compute

where we have chosen $\varepsilon_1 = \varepsilon_1(p, c_*, M, \vartheta) > 0$ sufficiently small. Now we prove the second inequality in (4.1). The statement is obvious for $h = 0$. If $h > 0$ and (4.1) holds, we have that

by our choice of $\vartheta$ and $\varepsilon_1$. Thus thanks to (4.2) we can apply Proposition 3.1 with $\vartheta^h r$ in place of $r$ to deduce that

where we have chosen $\vartheta_0 = \vartheta_0(c_*, \alpha)$ sufficiently small and we have used (4.3). Therefore, the second inequality in (4.1) is also true for every $k\in \mathbb{N}$. □

Analogously, it is possible to prove an iteration lemma for $U_{**}$.

Lemma 4.2. Let $(u, E)$ be a local minimizer of the functional $\mathcal{I}$ and let $\beta$ be the exponent of Proposition 3.2. For every $M > 0$ and $\vartheta\in (0, \vartheta_0)$, with $\vartheta_0 < \min\left\{ c_{**}, \frac{1}{4}\right\}$, there exist $\varepsilon_1 > 0$ and $R > 0$ such that, if $r < R$ and $x_0\in\Omega$ satisfy

where $c_{**}$ is the constant introduced in Proposition 3.2, then

Proof of Theorem 1.3. We consider the set

and let $x_0\in \Omega_1$. For every $M > 0$ and for $\varepsilon_1$ determined in Lemma 4.1 there exists a radius $R_{M, \varepsilon_1} > 0$ such that

for every $0 < r < R_{M, \varepsilon_1}$. Let $0 < \rho < \vartheta r < R$ and $h\in\mathbb{N}$ be such that $\vartheta^{h+1}r < \rho < \vartheta^h r$, where $\vartheta = \frac{\vartheta_0}{2}$ and $\vartheta_0$ is the same constant appearing in Lemma 4.1. By Lemma 4.1, we obtain

Using the properties of Lemma 2.2 and reasoning as in the proof of Lemma 4.1, we estimate

Thus, taking the previous chain of inequalities into account, applying again Lemma 4.1, we estimate

The previous estimate implies that

where $C_{*} = C_{*}(n, p, c_*, M, \vartheta_0)$. Since $U_*(y, r)$ is continuous in $y$, we have that $U_*(y, r) < \varepsilon_1$ for every $y$ in a suitable neighborhood $I$ of $x_0$. Therefore, for every $y\in I$ we have that

The last inequality implies, by the Campanato characterization of Hölder continuous functions (see [32, Theorem 2.9]), that $u$ is $C^{1, \alpha}$ in $I$ for every $0 < \alpha < \frac{1}{2}$, and we can conclude that the set $\Omega_1$ is open and the function $u$ has Hölder continuous derivatives in $\Omega_1$.

When the assumption (H) is not enforced, the proof goes exactly in the same way provided we use Lemma 4.2 in place of Lemma 4.1, with

□

5.   Conclusions

In this paper, we studied the $C^{1, \alpha}$ partial regularity for a wide class of multidimensional vectorial variational problems involving both bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with subquadratic growth $p\in (1, 2)$. Since the case $p \geq 2$ had been addressed in a previous work by the authors, the present paper completes the analysis by covering the entire range $p > 1$. The overall strategy of the proof is to establish an excess decay property for a suitably chosen excess function. The core of the argument - and the main contribution of the paper - is Proposition 3.1, where a one-step improvement of the excess is established. The proof proceeds via a contradiction and blow-up argument. The proof of Proposition 3.1 is rather long; nevertheless, we would like to highlight two fundamental estimates that are pivotal in the proof strategy. These are the Caccioppoli estimate (3.16) and the higher integrability estimate (3.47) for the blow-up sequences, in which the influence of the set $E$ appears explicitly. These estimates, together with the Sobolev–Poincaré inequality (2.7), which is specific to the subquadratic case, constitute the main tools used to establish the result.

Finally, we would like to mention two possible directions for future research, kindly suggested by one of the referees. The first concerns the potential extension of the same type of regularity to the non-uniformly elliptic case. Another intriguing question concerns the double-phase case, which may be more challenging, but should still be manageable - at least in the situation where the two phases are separated in the sets $E$ and $\Omega \setminus E$.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and wish to acknowledge financial support from INdAM GNAMPA Project 2024 "Regolarità per problemi a frontiera libera e disuguaglianze funzionali in contesto finsleriano" (CUP E53C23001670001).

Conflict of interest

Prof. Menita Carozza is a Guest Editor of special issue ''Multi-Rate Processes and Hysteresis" for Mathematics in Engineering. Prof. Menita Carozza was not involved in the editorial review and the decision to publish this article.

The author declares no conflicts of interest.