Moving surfaces and interfaces : application to damage, fracture and wearcontact

1.   Introduction

In this article we consider the following quasilinear boundary value problem

where $\Omega \subset \mathbb{R}^n$, with $n \geq 3$, $f, u: \Omega \to \mathbb{R}^N$, with $N \geq 2$, and $a: \Omega \times \mathbb{R}^N \to \mathbb{R}^{N^2 n^2}$ is a matrix valued function whose entries are $a_{i, j}^{\alpha, \beta} (x, u)$ with $i, j \in \{1, \ldots, n\}$ and $\alpha, \beta \in \{1, \ldots, N\}$. Therefore the first line in (1.1) is a system of $N$ equations of the form

For the treated problem there is an extensive literature in the scalar case $N = 1$.

In particular, for the existence of a suitably defined solution, the Reader can refer to the papers ^{[9,10,14,58,69]} while, relatively to uniqueness and a priori estimates, we can quote respectively the papers ^[67] and ^[3]. For what concerns the regularity of a solution we cite the works ^[35,38]. Moreover, similar conclusions for the nonlinear case can be found in ^[2,8,12] and for the anisotropic case in ^[4,36]. Subsequently the aforementioned results have been extended to the operator with lower order terms too (see also ^{[15,16,17,20,51]}). In this context one can also see ^{[7,11,12,24,25,30]}. Furthermore, in ^[37] the right hand side appears in divergence form, that is $f = - {\rm{div}} F$ and in ^[1] the biharmonic operator is studied.

For further regularity results concerning elliptic operators the Reader is invited to refer to the foundamental works ^{[5,6,26,27,28,41,42,43,44,45,46,47,48,64]} and the survey ^[65].

As it is shown by the De Giorgi's counterexample ^[29], see also ^{[39,40,49,60,61]}, the good regularity properties obtained in the scalar case can not be in general extended to the vectorial one, unless new structural assumptions are introduced.

An existence result of bounded weak solution for nonlinear degenerate elliptic systems is obtained in ^[55], using a componentwise coercivity condition. In several other papers, conditions on the support of the off-diagonal coefficients $a^{\alpha, \beta}_{i, j}(x)$ have been used to address different problems. Let us mention that a maximum principle result is obtained in ^[66] where the assumption is $a_{i, j}^{\alpha, \beta}(x, y) = 0$ for $\alpha \ne \beta$ when $y^\alpha$ is large and in ^[52] where different shapes of support are considered. Hölder continuity of the solutions is proved in ^[70] for a tridiagonal system, $a^{\alpha, \beta}_{i, j} = 0$ for $\beta > \alpha$. $L^{\infty}$ regularity results are obtained in ^[53] for an oblique type of support for the coefficients and in ^[54] for a butterfly support. Measure data problems are faced in ^[56] and ^[57] where the support of $a_{i, j}^{\alpha, \beta}(x, y)$ is contained in squares along the $y^\alpha = \pm y^\beta$ diagonals.

These kind of assumptions on the coefficients have been recently employed also to deal with degenerate elliptic systems. In this context there are results on problem (1.1) when the datum $f \in L^m$, which extend the ones contained in ^[14] for the scalar case. Namely, in ^[31] the existence of a bounded solution is proved when $m > \frac{n}{2}$, assuming a butterfly support for the off-diagonal coefficients; moreover in ^[32] the case of a datum $f$ with an intermediate grade of integrability ($m < \frac{n}{2}$) is treated, thanks to an appropriate choice of the support for the off-diagonal coefficient.

In this paper we extend to the degenerate vectorial problem (1.1) an existence result concerning degenerate scalar operators, with the datum $f$ in a suitable Marcinkiewicz space, contained in ^[14,62,63] (see also ^[21,22]). Since we are dealing with the vectorial case the support of the coefficients is required to have a particular structure. In Section 2 we give the precise notions of degenerate ellipticity and Marcinkiewicz spaces, see respectively $(\mathcal{A}_2)$ and definition 1, while the assumption on the shape of the support of the coefficients is stated in ($\mathcal{A}_3$).

Also in this context the extension to the vectorial case of the known result in the scalar one is not obvious. Indeed, starting from De Giorgi's counterexample, it is possible to construct an example of an elliptic system with datum $f \in L^p$ for every $p < n$, whose unique solution is unbounded and has low integrability, see ^[31] for details on the counterexample.

When dealing with systems of $N$ equations, like (1.2), whose coefficients are only measurable with respect to $x$, little is known. Most articles are devoted to study existence or regularity of solutions of systems with right hand side $f^\alpha \in L^m$, either when $m$ is large, or when $m$ is small. When $m$ is large, namely $m > \frac{n}{2}$, existence of bounded solutions is obtained in ^[31].

When $m$ is small, namely $m = 1$, or even when $f^\alpha$ is a measure, existence of solutions have been studied for general systems

under structure conditions on $A^{\alpha}_{i}$. Namely, in ^[33] and ^[34], authors assume that

for every $b \in \mathbb{R}^N$ with $|b| \leq 1$. On the other hand, in ^[71], the author assumes the componentwise sign condition

for every $\alpha = 1, \dots, N$. When $N = 2$, (1.4) implies (1.5): it is enough to take first $b = (1, 0)$, then $b = (0, 1)$. Note that, in the present paper, we address the quasilinear case

in this case, as far as one off-diagonal coefficient $a_{\tilde{i}, \tilde{j}}^{\tilde{\alpha}, \tilde{\beta}}(x, y)$ is non zero, then (1.5) is no longer true: it is enough to take $\alpha = \tilde{\alpha}$, $\xi^{\beta}_{j} = 0$ if $\beta \notin \{ \tilde{\alpha}, \tilde{\beta} \}$, $\xi^{\tilde{\alpha}}_{i} = 0$ if $i \neq \tilde{i}$, $\xi^{\tilde{\alpha}}_{\tilde{i}} = 1$, $\xi^{\tilde{\beta}}_{j} = 0$ if $j \neq \tilde{j}$, $\xi^{\tilde{\beta}}_{\tilde{j}} = t \frac{a_{\tilde{i}, \tilde{j}}^{\tilde{\alpha}, \tilde{\beta}}(x, y)}{\left| a_{\tilde{i}, \tilde{j}}^{\tilde{\alpha}, \tilde{\beta}}(x, y) \right|^2}$ with $t \to -\infty$. When $N = 2$, failure of (1.5) implies failure of (1.4). We recall that the study of quasilinear systems (1.2) with $f^\alpha \in L^1$ is contained in ^[57] under the assumption that the support of off-diagonal coefficients is contained in a sequence of squares with side lenght $r$ along the diagonals of the $y^{\alpha}-y^{\beta}$ plane.

Concerning existence and regularity of suitable defined solutions of linear ellitptic systems

with VMO coefficients and datum $f = (f^{\alpha})$ in a Lebesgue space $L^{\gamma}$ with $\gamma \in (1, \frac{2n}{n+2}]$ (i.e., below the duality exponent) or in a suitable Lorentz-Morrey space one can refer also to ^[50]. While if $f$ belongs to the natural dual Lebesgue space but the linear operator in not coercive due to the presence of a lower order term, called "drift term",

then existence and regularity results can be found in ^[19]. The above result has been extended to non linear operator under the so-called Landes condition (similar to (1.4)) with datum being in $L^1$ or in $L^{\frac{2n}{n+2}}$ respectively in ^[18,23].

In the present work we address the existence of a regular solution to (1.2) when $f^\alpha$ has an intermediate degree of integrability, namely, $f^\alpha \in M^m$ with $1 < \frac{2n}{n + 2 - \theta (n-2)} < m < \frac{n}{2}$ and $M^m$ is the Marcinkiewicz space. In this case, the higher degree of integrability of the right hand side $f^\alpha$ allows us to weaken the condition on the support of off-diagonal coefficients.

In the next section we present assumptions on the coefficients and on the datum $f$ and our result. In Section 3 we consider a sequence of approximating non degenerate problems and we prove estimates on their weak solutions; then, with a limit procedure, we get the result for our problem.

2.   Assumptions and result

For all $i, j \in \{1, \ldots, n\}$ and all $\alpha, \beta \in \{1, \ldots, N\}$ we assume that $a_{i, j}^{\alpha, \beta}: \Omega \times \mathbb{R}^N \to \mathbb{R}$ satisfies the following conditions:

($\mathcal{A}_0$) $x \mapsto a_{i, j}^{\alpha, \beta}(x, y)$ is measurable and $y \mapsto a_{i, j}^{\alpha, \beta}(x, y)$ is continuous;

($\mathcal{A}_1$) (boundedness of all the coefficients) there exists $c > 0$ such that

for almost every $x \in \Omega$ and for all $y \in \mathbb{R}^N$;

($\mathcal{A}_2$) (degenerate ellipticity of all the coefficients) there exist constants $\nu > 0$ and $\theta \in (0, 1)$ such that

for almost every $x \in \Omega$, for all $y \in \mathbb{R}^N$ and $\xi \in \mathbb{R}^{N \times n}$;

($\mathcal{A}_3$) (support of off-diagonal coefficients) there exists $L_0 \geq 1$ such that $(\mathcal{A'}_3)$ and $(\mathcal{A''}_3)$ hold, where

($\mathcal{A'}_3$) (support of off-diagonal coefficients contained in a central square) if $a_{i, j}^{\alpha, \beta}(x, y) \neq 0$ and $0 \le |y^\alpha| < L_0$, then it holds also $0 \le |y^\beta| < L_0$;

($\mathcal{A''}_3$) (support of off-diagonal coefficients contained in the union of a geometric progression of squares) if $a_{i, j}^{\alpha, \beta}(x, y) \neq 0$ and there exists $t \in \mathbb{N} \cup \{0\}$ such that $2^t L_0 \leq |y^\alpha| < 2^{t+1} L_0$, then it holds also $2^t L_0 \leq |y^\beta| < 2^{t+1} L_0$.

Let us remark that from assumption $(\mathcal{A}_2)$ it follows that we have degeneracy in the $\alpha$ equation when $u^{\alpha}$ is large. In ^[13] is treated for $N = 2$ the case in which degeneracy in the $\alpha$ equation arises when $u^{\beta}$ is large, with $\beta \neq \alpha$.

Note that ($\mathcal{A'}_3$) and ($\mathcal{A''}_3)$ are always fulfilled when $\alpha = \beta$. On the contrary, when $\alpha \neq \beta$, ($\mathcal{A}_3$) forces the support of $a_{i, j}^{\alpha, \beta}(x, y)$ to be contained in the union of infinite squares along the diagonals, see grey region in Figure 1.

On $f$ we assume that it belongs to the Marcinkiewicz space $M^m(\Omega, \mathbb{R}^N)$, with

For the convenience of the Reader, we recall the definition of Marcinkiewicz spaces, also known as weak Lebesgue spaces.

Definition 2.1. Let $m$ be a positive number. We say that a measurable function $f: \Omega \to \mathbb{R}$ belongs to the Marcinkiewicz space $M^m(\Omega, \mathbb{R})$ if there exists a positive constant $c$ sucht that

in such a case we set

$M^m(\Omega, \mathbb{R}^N)$ is the space of functions $f = (f^1, \ldots, f^N)$ such that $f^i \in M^m(\Omega, \mathbb{R})$ for each $i$. Moreover $M_m(f) = \sum_{\alpha = 1}^N M_m(f^{\alpha})$.

We recall some properties on Marcinkiewicz spaces:

and

For more details on Marcinkiewicz space see ^[10,68].

Let us explicitly remark that, being $0 < \theta < 1$, from (2.2) it follows that

Under our set of assumptions we prove the following theorem:

Theorem 2.1. Assume $(\mathcal{A}_0)$, $(\mathcal{A}_1)$, $(\mathcal{A}_2)$, $(\mathcal{A}_3)$, with $n \geq 3$. If $f \in M^m(\Omega, \mathbb{R}^N)$, with $\frac{2n}{n+2-\theta(n-2)} < m < \frac{n}{2}$, then there exists $u \in W_0^{1, 2}(\Omega, \mathbb{R}^N) \cap M^r(\Omega, \mathbb{R}^N)$, with

weak solution of the problem (1.1), that is such that

for all $\varphi \in W_0^{1, 2}(\Omega, \mathbb{R}^N)$.

3.   Approximation and estimates

We set for all $k \in \mathbb{N}$

with

We consider the following family of approximating problems

We want to show the existence of a weak solution for each problem ($\mathcal{\tilde{P}}_k$), that is a function $u_k \in W_0^{1, 2}(\Omega, \mathbb{R}^N)$ such that

for all $\varphi \in W_0^{1, 2}(\Omega, \mathbb{R}^N)$.

Let us first show some properties of the coefficients $\tilde{a}_{i, j, k}^{\alpha, \beta}$. From assumption ($\mathcal{A}_1$) it follows that

Using assumption ($\mathcal{A}_2$) we have the following non degenerate ellipticity condition

Now let us show that for all $f \in M^m(\Omega, \mathbb{R}^N)$, with $m > \frac{2n}{n+2-\theta(n-2)}$, the linear operator

is continuous. Indeed, using Hölder inequality, (2.4) and Sobolev embedding, we have for a suitable constant $C > 0$

and the continuity of $F$ is proved. Therefore we can apply the surjectivity result of Leray-Lions, see ^[59], and we have the existence of a weak solution $u_k$ for the problem $(\mathcal{\tilde{P}}_k)$, that is, there exists $u_k \in W^{1, 2}_0(\Omega, \mathbb{R}^N)$ such that (3.1) holds true for every $\varphi \in W^{1, 2}_0(\Omega, \mathbb{R}^N)$.

In the next Lemma 3.1, arguing as in ^[14], we prove that the sequence $\{u_k\}_{k \in \mathbb{N}}$ is bounded in $W_0^{1, 2}(\Omega, \mathbb{R}^N) \cap M^r(\Omega, \mathbb{R}^N)$. We first recall the following elementary inequalities that will be used in the proof of Lemma 3.1. We have

provided $a_{\alpha}, b_{\alpha} \geq 0$ for all $\alpha \in \{ 1, \dots, M\}$ and $p > 0$.

Lemma 3.1. Assume that $f \in M^m(\Omega, \mathbb{R}^N)$ with $\frac{2n}{n+2-\theta(n-2)} < m < \frac{n}{2}$ and let $u_k$ be a weak solution of $(\mathcal{\tilde{P}}_k)$. Then the sequences $\|u_k\|_{W_0^{1, 2}(\Omega, \mathbb{R}^N)}$ and $M_r(u_k, \Omega)$, with $r$ given in (2.5), are bounded by a positive constant which depends only on $L_0$, $\theta$, $m$, $n$, $N$, $\nu$, $|\Omega|$ and $M_m(f, \Omega)$.

Proof. For any $t \in \mathbb{N} \cup \{0\}$ and for $L_0 \geq 1$ given by assumption $(\mathcal{A}_3)$, we define the following functions

and

We consider as test function in (3.1) the function $\varphi_t \in W_0^{1, 2}(\Omega, \mathbb{R}^N)$ defined as

We introduce the sets

For all $\alpha \in \{1, \ldots, N\}$ we have

where $\mathbb{I}_B(x) = 1$ if $x \in B$ and $\mathbb{I}_B(x) = 0$ otherwise. Moreover, using $(\mathcal{A''}_3)$, we have

Indeed, if $a_{i, j}^{\alpha, \beta} (x, u_k(x)) = 0$ or $x\notin B_{k, 2^t L_0}^{\alpha}$, then the (3.7) is obvious. If $a_{i, j}^{\alpha, \beta} (x, u_k(x)) \neq 0$ and $x\in B_{k, 2^t L_0}^{\alpha}$, that is $2^t L_0\le \vert u_k^{\alpha} \vert < 2^{t+1} L_0$, then for $(\mathcal{A''}_3)$ we have $2^t L_0 \le \vert u_k^{\beta} \vert < 2^{t+1} L_0$ so that $x\in B_{k, 2^t L_0}^{\beta}$.

From (3.6), (3.7) and $(\mathcal{A}_2)$ we have

Then, replacing in the left side of (3.1) the test function (3.5) and using (3.8), we get

Combining (3.9) with (3.1), we get

Using Sobolev's embedding and (3.10) we have

where $C_S$ is the Sobolev embedding constant. Summing on $\alpha$ in (3.11) and using (3.10), we have

From (2.4) we have $f \in L^{\frac{2n}{n+2}}(\Omega)$ and, by Sobolev immersion, we have also $\varphi_t \in L^{2^{\ast}} = L^{\frac{2n}{n-2}}$. Then, using the Hölder inequality with exponents $\frac{2n}{n+2}$ and $\frac{2n}{n-2} = 2^{\ast}$ and applying (2.3) to the function $|f^{\alpha}|^{\frac{2n}{n+2}} \in M^{\frac{(n+2)m}{2n}}(\Omega)$, we deduce for all $\alpha = 1, \ldots, n$

From (3.12) and (3.13) it follows that

where $C_1$ is a constant depending only on $C_s$, $\nu$, $M_m(f, \Omega)$.

Now, using last inequality and (3.3), (3.4), (3.2), we have

Therefore

Since $\varphi_t^{\alpha} = T_{2^t L_0} (G_{2^t L_0}(u_k^{\alpha}))$, for all $t \in \mathbb{N} \cup \{0\}$, we have

Then, summing on $\alpha$ and using (3.2) we have

From (3.14) and (3.15) it follows that

and then

From the last inequality and using (3.3) and (3.2) we have

Therefore

where $C_2$ is a positive constant depending only on $N$, $n$, $C_S$, $\nu$ and $M_m(f, \Omega)$.

Let us set

and let us remark that for $r$ given in (2.5) the following equality holds

Now, for all $h > 0$ and for $r$ given in (2.5), let us define

For all $t \in \mathbb{N} \cup \{0\}$, it follows from (3.16) that

Therefore we obtain that there exists a constant $C_3 = \max (1, 2^r C_2) \geq 1$, depending only on $N$, $n$, $C_S$, $\nu$, $M_m(f, \Omega)$, $\theta$ and $m$ such that, for all $t \in \mathbb{N} \cup \{0\}$, we have

and, arguing by induction, it follows that

Being $\gamma < 1$ and $\rho(L_0) \geq 0$, the elementary inequality

holds. Using the notation $C_4 = C_3^{\sum_{h = 0}^{+\infty}\gamma^h} = C_3^{\frac{1}{1-\gamma}}$ and putting together (3.19) and (3.20), we have

where $C_4 \geq 1$ is a constant depending only on $N$, $n$, $C_S$, $\nu$, $M_m(f, \Omega)$, $\theta$ and $m$.

Using (3.21), we want to prove that there exists a constant $C_5$ depending only on $N$, $n$, $C_S$, $\nu$, $M_m(f, \Omega)$, $\theta$, $m$, $L_0$ and $|\Omega|$ such that

Indeed, for $h \in [L_0, 2 L_0]$, we have

For all $h \geq 2 L_0$ there exists $s \in \mathbb{N}$ and $w \in [L_0, 2 L_0)$ such that $h = 2^s w$. Then, using (3.21) and (3.23), we have for all $h \geq 2 L_0$

From (3.23) and (3.24) follows (3.22).

For all $h \geq L_0$, using (3.22), we have

for $h \in (0, L_0)$ we have

Then, setting $C_6 = \max (C_5, N |\Omega|L_0^r)$, from (3.25) and (3.26), we get

proving the boundness of the sequence $\{u_k\}_{k \in \mathbb{N}}$ in $M^r(\Omega)$.

It remains to prove that the sequence $\{u_k\}_{k \in \mathbb{N}}$ is bounded in $W^{1, 2}_0(\Omega, \mathbb R^N).$

From (3.10), for all $t\in \mathbb N\cup \{0\}$, we have

Now we estimate the right hand side of (3.27).

Observing that $\vert \varphi_t^{\alpha}(x) \vert\le 2^tL_0$, for all $x$ and for all $t\in \mathbb{N}\cup \{0 \}$, and using (2.3), we have

Summing on $\alpha = 1, .., N$ the previous inequality, by (3.2), the definition of $\rho$ in (3.18) and (3.22), we get

From this inequality it follows that

Since $\frac{2n}{n+2-\theta (n-2)} < m < \frac{n}{2}$, it results that $\theta +1-r(1-\frac{1}{m}) < 0$ and the series in the right side of the last inequality converges; we have

where $C_7$ is a positive constant depending only on $n, N, m, \theta, \nu, M_m(f, \Omega), \vert \Omega \vert, C_S, L_0.$

Now, let us prove that $\sum_{\alpha = 1}^N \int _{\{ \vert u_k^{\alpha} \vert < L_0 \} } \vert Du_k^{\alpha}(x)\vert^2dx$ is bounded.

To this aim we use $\psi = \big(\psi^1, ..., \psi^N\big) = \big(T_{L_0}(u_k^1), ..., T_{L_0}(u_k^N)\big)$ as a test function in the weakly formulation (3.1) of problem $(\tilde P_k)$. Observing that

where $B_{k, 0}^{\alpha} = \{ x\in \Omega : 0\le \vert u_k^{\alpha}(x)\vert < L_0\}$, we have

From $(\mathcal A'_3)$, we get

Combining (3.30), (3.31) and $(\mathcal A_2)$, we deduce that

From (3.32) and (2.3), it follows that

Therefore, combining (3.27), (3.29) and (3.33), we have

where $C_8$ is a positive constant depending only on $n, N, m, \theta, \nu, M_m(f, \Omega), \vert \Omega \vert, C_S, L_0,$ and the boundedness of $u_k$ in $W_0^{1, 2}(\Omega, \mathbb{R^N})$ is proved.

Proof of Theorem 2.1

Proof. Let $u_k$ be a solution of $(\tilde {P}_k)$. Lemma 3.1 states that the sequence of $\{u_k \}$ is uniformly bounded in $M^r(\Omega, \mathbb{R}^N)$ and in $W^{1, 2}_0(\Omega, \mathbb{R}^N)$. Then there exists a positive constant $C$ such that $M_r(u_k, \Omega) \leq C$ and $\|u_k\|_{W^{1, 2}_0(\Omega, \mathbb{R}^N)} \leq C$ for all $k \in \mathbb{N}$. Being $\{u_k \}$ bounded in $W^{1, 2}_0(\Omega, \mathbb{R}^N)$ there exists a subsequence $\{u_{k_{\lambda}} \}$ weakly converging in $W^{1, 2}_0(\Omega, \mathbb{R}^N)$ to a function $u \in W^{1, 2}_0(\Omega, \mathbb{R}^N)$. Moreover, by Rellich-Kondrachov embedding Theorem, Sobolev space $W_0^{1, 2}(\Omega, \mathbb{R}^N)$ is compactly embedded in $L^2(\Omega, \mathbb{R}^N)$; then, there exists a subsequence, not relabeled, also in the sequel, strongly converging to $u$ in $L^{2}$. From $L^2$ convergence we get pointwise convergence almost everywhere, up to a further subsequence. Briefly we write

Now, we pass to the limit as $\lambda \rightarrow +\infty$, in the weak formulation of problem $(\tilde {P}_k)$, written when $k = k_{\lambda}$, to prove that $u$ solves problem (2.6). More precisely, we verify that for all $\varphi \in W_0^{1, 2}(\Omega, \mathbb{R}^N)$

To this aim, we estimate

We obtain the result by proving that $I_{k_{\lambda}}, II_{k_{\lambda}}, III_{k_{\lambda}}$ tend to zero as $\lambda \rightarrow +\infty.$ We start to estimate $I_{k_{\lambda}}.$ Using Hölder inequality and boundedness of the sequence $\{u_{k_{\lambda}}\}$ in $W^{1, 2}_0(\Omega)$ we have

For any $i, j = 1, \ldots, n$ and for any $\alpha, \beta = 1, \ldots, N$, using pointwise convergence in (3.34) and continuity of functions $y \to a_{i, j}^{\alpha, \beta}(x, y)$ we have that

moreover from $(\mathcal{A}_1)$ we get

therefore, by dominated convergence theorem, we obtain that

The above limit and (3.35) imply that $I_{k_{\lambda}}$ tends to zero as $\lambda \to +\infty$.

Observing that $u_{k_{\lambda}}^{\beta} \rightharpoonup u^{\beta}$ in $W^{1, 2}_0(\Omega)$, for any $i, j = 1, \ldots, n$ and for any $\alpha, \beta = 1, \ldots, N$, we have

hence $II_{k_{\lambda}}$ tends to zero as $\lambda \to +\infty$.

Using Hölder inequality and (3.34), it results that

Passing to the limit as $\lambda \rightarrow +\infty$, we obtain that $III_{k_{\lambda}}$ tends to zero and the proof is completed.

Acknowledgments

We acknowledge the support of GNAMPA, INdAM, MIUR, UNⅣAQ. S. Leonardi has been supported by Piano della Ricerca di Ateneo 2020-2022–PIACERI: Project MO.S.A.I.C. "Monitoraggio satellitare, modellazioni matematiche e soluzioni architettoniche e urbane per lo studio, la previsione e la mitigazione delle isole di calore urbano", Project EEEP & DLaD. S. Leonardi is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), codice CUP_E55F22000270001.

Conflict of interest

The authors declare no conflict of interest.