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Small solids in an inviscid fluid

  • We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.

    Citation: Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Small solids in an inviscid fluid[J]. Networks and Heterogeneous Media, 2010, 5(3): 385-404. doi: 10.3934/nhm.2010.5.385

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  • We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.


    We consider the general second order elliptic equation in divergence form

    ni=1xiai(x,u(x),Du(x))=b(x,u(x),Du(x)),xΩ, (1.1)

    where Ω is an open set of Rn, n2, the vector field (ai(x,u,ξ))i=1,,n and the right hand side b(x,u,ξ) are Carathéodory applications defined in Ω×R×Rn. We study the elliptic equations (1.1) under some general growth conditions on the gradient variable ξ=Du, named p,q conditions, which we are going to state in the next Section 3.2. Under these assumptions we will obtain the local boundedness of the weak solutions, as stated in Theorem 3.2.

    A strong motivation to study the local boundedness of solutions to (1.1) relies on the recent research in [53], where the local Lipschitz continuity of the weak solutions of the Eq (1.1) has been obtained under general growth conditions, precisely some p,qgrowth assumptions, with the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable. In [53] the Sobolev class of functions where to start in order to get more regularity of the weak solutions was pointed out, precisely uW1,qloc(Ω)Lloc(Ω). That is, in particular the local boundedness uLloc(Ω) of weak solutions is a starting assumption for more interior regularity; i.e., for obtaining uW1,loc(Ω) and more. When we refer to the classical cases this is a well known aspect which appears in the mathematical literature on a-priori regularity: in fact, for instance, under the so-called natural growth conditions, i.e., when q=p, then the a-priori boundedness of u often is a natural assumption to obtain the boundedness of its gradient Du too; see for instance the classical reference book by Ladyzhenskaya-Ural'tseva [45,Chapter 4,Section 3] and the C1,αregularity result by Tolksdorf [60].

    The aim of this paper is to derive the local boundedness of solutions to (1.1); i.e., to deduce the local boundedness of u only from the growth assumptions on the vector field (ai(x,u,ξ))i=1,,n and the right hand side b(x,u,ξ) in (1.1). The precise conditions and the related results are stated in Section 3.

    We start with a relevant aspect to remark in our context, which is different from what happens in minimization problems and it is peculiar for equations: although under p,qgrowth conditions (with p<q) the Eq (1.1) is elliptic and coercive in W1,ploc(Ω), it is not possible a-priori to look for weak solutions only in the Sobolev class W1,ploc(Ω), but it is necessary to emphasize that the notion of weak solution is consistent if a-priori we assume uW1,qloc(Ω). This is detailed in Section 2.

    Going into more detail, in this article we study the local boundedness of weak solutions to the pelliptic equation (1.1) with qgrowth, 1<pq<p+1, as in (3.2), (3.3) and (3.7)–(3.10). Starting from the integrability condition uW1,qloc(Ω) on the weak solution, under the bound on the ratio qp

    qp<1+1n1

    we obtain uLloc(Ω). The proof is based on the powerful De Giorgi technique [29], by showing first a Caccioppoli-type inequality and then applying an iteration procedure. The result is obtained via a Sobolev embedding theorem on spheres, a procedure introduced by Bella and Schäffner in [3], that allows a dimensional gain in the gap between p and q. This idea has been later used by the same authors in [4], by Schäffner [58] and, particularly close to the topic of our paper, by Hirsch and Schäffner [43] and De Rosa and Grimaldi [30], where the local boundedness of scalar minimizers of a class of convex energy integrals with p,qgrowth was obtained with the bound qp<1+qn1.

    Some references about the local boundedness of solutions to elliptic equations and systems, with general and p,qgrowth conditions, start by Kolodīĭ [44] in 1970 in the specific case of some anisotropic elliptic equations. The local boundedness of solution to classes of anisotropic elliptic equations or systems have been investigated by the authors [18,19,20,21,22,23,24] and by Di Benedetto, Gianazza and Vespri [31]. Other results on the boundedness of solutions of PDEs or of minimizers of integral functionals can be found in Boccardo, Marcellini and Sbordone [7], Fusco and Sbordone [37,38], Stroffolini [59], Cianchi [14], Pucci and Servadei [57], Cupini, Leonetti and Mascolo [17], Carozza, Gao, Giova and Leonetti [12], Granucci and Randolfi [42], Biagi, Cupini and Mascolo [5].

    Interior Lgradient bound, i.e., the local Lipschitz continuity, of weak solutions to nonlinear elliptic equations and systems under non standard growth conditions have been obtained since 1989 in [46,47,48,49,50]. See also the following recent references for other Lipschitz regularity results: Colombo and Mingione [16], Baroni, Colombo and Mingione [1], Eleuteri, Marcellini and Mascolo [34,35], Di Marco and Marcellini [32], Beck and Mingione [2], Bousquet and Brasco [9], De Filippis and Mingione [26,27], Caselli, Eleuteri and Passarelli di Napoli [13], Gentile [39], the authors and Passarelli di Napoli [25], Eleuteri, Marcellini, Mascolo and Perrotta [36]; see also [53]. For other related results see also Byun and Oh [10] and Mingione and Palatucci [55]. The local boundedness of the solution u can be used to achieve further regularity properties, as the Hölder continuity of u or of its gradient Du; we limit here to cite Bildhauer and Fuchs [6], Düzgun, Marcellini and Vespri [33], Di Benedetto, Gianazza and Vespri [31], Byun and Oh [11] as examples of this approach. For recent boundary regularity results in the context considered in this manuscript we mention Cianchi and Maz'ya [15], Bögelein, Duzaar, Marcellini and Scheven [8], De Filippis and Piccinini [28]. A well known reference about the regularity theory is the article [54] by Giuseppe Mingione. We also refer to [51,52,53] and to De Filippis and Mingione [27], Mingione and Rădulescu [56], who have outlined the recent trends and advances in the regularity theory for variational problems with non-standard growths and non-uniform ellipticity.

    In order to investigate the consistency of the notion of weak solution, we anticipate the ellipticity and growth conditions of Section 3, in particular the growth in (3.3), (3.4),

    {|ai(x,u,ξ)|Λ{|ξ|q1+|u|γ1+b1(x)},i=1,,n,|b(x,u,ξ)|Λ{|ξ|r+|u|γ2+b2(x)}. (2.1)

    As well known the integral form of the equation, for a smooth test function φ with compact support in Ω, is

    Ωni=1ai(x,u,Du)φxidx+Ωb(x,u,Du)φdx=0.

    Let us discuss the summability conditions for the pairings above to be well defined. Since each ai in the gradient variable ξ grows at most as |ξ|q1, more generally we can consider test functions φW1,q0(Ω). In fact, starting with the first addendum and applying the Young inequality with conjugate exponents qq1 and q, we obtain the L1 local summability

    |ai(x,u,Du)φxi|Λ{|Du|q1+|u|γ1+b1(x)}|φxi|Λq1q{|Du|q1+|u|γ1+b1(x)}qq1+Λq|φxi|qL1loc(Ω)

    if uW1,qloc(Ω) and if qq1γ1q, where q is the Sobolev conjugate exponent of q, and b1Lqq1loc(Ω). On γ1 equivalently we require (if q<n) γ1qq1q=nqnqq1q=n(q1)nq, which essentially corresponds to our assumption (3.8) below (the difference being the strict sign "<" for compactness reasons). We also observe that the summability condition b1Lqq1loc(Ω) is satisfied if b1Ls1loc(Ω), with s1>nq1, as in (3.10).

    Similar computations apply to |b(x,u,ξ)φ|, again if q<n and with conjugate exponents qq1 and q,

    |b(x,u,Du)φ|Λ{|Du|r+|u|γ2+b2(x)}|φ|Λq1q{|Du|r+|u|γ2+b2(x)}qq1+Λq|φ|qL1loc(Ω)

    and we obtain b2Lqq1loc(Ω) (compare with (3.10), where b2Ls2loc(Ω) with s2>np, since qq1pp1ppp=np) and the conditions for r and γ2 expressed by rqq1q and γ2qq1q; i.e., for the first one,

    rqq1q=qnqnq1nqnq=q+qn1,

    which correspond to the more strict assumption (3.9), with r<p+pn1, with the sign "<" and where q is replaced by p. Finally for γ2 we obtain γ2q1, which again corresponds to our assumption (3.8) with the strict sign.

    Therefore our assumptions for Theorem 3.2 are more strict than that ones considered in this section and they are consistent with a correct definition of weak solution to the elliptic equation (1.1).

    Let ai:Ω×R×RnR, i=1,...,n, and b:Ω×R×RnR be Carathéodory functions, Ω be an open set in Rn, n2. Consider the nonlinear partial differential equation

    ni=1xiai(x,u,Du)=b(x,u,Du). (3.1)

    For the sake of simplicity we use the following notation: a(x,u,ξ)=(ai(x,u,ξ))i=1,...,n, for all i=1,,n.

    We assume the following properties:

    pellipticity condition at infinity:

    there exist an exponent p>1 and a positive constant λ such that

    a(x,u,ξ),ξλ|ξ|p, (3.2)

    for a.e. xΩ, for every uR and for all ξRn such that |ξ|1.

    qgrowth condition:

    there exist exponents qp, γ10, s1>1, a positive constant Λ and a positive function b1Ls1loc(Ω) such that, for a.e. xΩ, for every uR and for all ξRn,

    |a(x,u,ξ)|Λ{|ξ|q1+|u|γ1+b1(x)}; (3.3)

    growth conditions for the right hand side b(x,u,ξ):

    there exist further exponents r0, γ20, s2>1 and a positive function b2Ls2loc(Ω) such that

    |b(x,u,ξ)|Λ{|ξ|r+|u|γ2+b2(x)}, (3.4)

    for a.e. xΩ, for every uR and for all ξRn.

    Without loss of generality we can assume Λ1 and b1,b21 a.e. in Ω. We recall the definition of weak solution to (3.1).

    Definition 3.1. A function uW1,qloc(Ω) is a weak solution to (3.1) if

    Ω{ni=1ai(x,u,Du)φxi+b(x,u,Du)φ}dx=0 (3.5)

    for all φW1,q(Ω), suppφΩ.

    Our aim is to study the local boundedness of weak solutions to (3.1). Since this regularity property is trivially satisfied for functions in W1,qloc(Ω) with q>n, from now on we only consider the case qn; more precisely

    1<p<n,pqn, (3.6)

    since if q>n then weak solutions are Hölder continuous as an application of the Sobolev-Morrey embedding theorem, see Remark 3.3.

    Other assumptions on the exponents are

    {q<1+pqp<1+1n1 (3.7)
    0γ1<n(q1)np,0γ2<n(p1)+pnp, (3.8)
    0r<p+pn1, (3.9)
    s1>nq1,s2>np. (3.10)

    Under the conditions described above the following local boundedness result holds.

    Theorem 3.2 (Boundedness result). Let uW1,qloc(Ω), 1<qn, be a weak solution to the elliptic equation (3.1). If (3.2)–(3.4) and (3.6)–(3.10) hold true, then u is locally bounded. Precisely, for every open set ΩΩ there exist constants R0,c>0 depending on the data n,p,q,r,γ1,γ2,s1,s2 and on the norm uW1,q(Ω) such that uL(BR/2(x0))c for every RR0, with BR0(x0)Ω.

    Remark 3.3. We already observed that if q>n then the weak solutions to (3.1) are locally Hölder continuous. Let us now discuss why in (3.6) we do not consider the case p=q=n. If p=q (n), the same computations in the proof of Theorem 3.2 work with the set of assumptions (3.8)–(3.10). They can be written, coherently with the previous ones, as

    0γ1<pp1p,0γ2<p1 (3.11)
    0r<ppp, (3.12)
    s1>pp(pp)(p1),s2>ppp. (3.13)

    Here p denotes the Sobolev exponent appearing in the Sobolev embedding theorem for functions in W1,p(Ω) with Ω bounded open set in Rn; i.e.,

    p:={npnp if p<nany real number >n, if p=n. (3.14)

    Following the computations in [40,Theorem 2.1] and [41,Chapter 6] it can be proved that the weak solutions to (3.1) are quasi-minima of the functional

    F(u):=Ω(|Du|p+|u|τ+bpp11+bpp12)dx, (3.15)

    with τ:=max{γ1pp1,γ2pp1}. It is known that if

    τ<pandbpp11+bpp12L1+δ  with δ>0 (3.16)

    then the gradient of quasi-minima of the functional (3.15) satisfies a higher integrability property; i.e., they belong to W1,p+ϵ, for some ϵ>0.

    Under our assumptions, (3.16) is satisfied; indeed, taking into account that we are considering p=q, by (3.10)

    s1>np1pp1

    and, by (3.13)

    s2>ppppp1.

    Analogously, by (3.11),

    γ1pp1<p,γ2pp1<(p1)pp1=p.

    In particular, if p=q=n the quasi-minima of (3.15) are in W1,n+ϵloc(Ω) for some ϵ>0, therefore the weak solutions to (3.1) are Hölder continuous. We refer to [41] Chapter 6 for more details.

    If p1 and dN, d2, we define

    (pd):={dpdp if p<dany real number >d, if p=d.

    The Sobolev exponent appearing in the Sobolev embedding theorem for functions in W1,p(Ω), p1, with Ω bounded open set in Rn, is (pn) and will be denoted, as usual, p.

    Let tR, t>0. We define t as follows:

    1t:=min{1t+1n1,1}.

    We have, if n3,

    t={t(n1)t+n1if t>n1n21if 1tn1n2,

    and, if n=2, t=1 for every t.

    We notice that, if n3,

    ((t)n1)={tif  t>n1n2n1n2if  1tn1n2

    and, if n=2, for every t, ((t)n1) stands for any real number greater than 1.

    Remark 4.1. Let us consider the exponents p,q satisfying (3.6) and (3.7) in Section 3. We notice that

    1(ppq+1)={1ppq+1+1n1if q>1+pn11if q1+pn1. (4.1)

    Due to assumption (3.7), if n=2, then (ppq+1)=1.

    Moreover, if we denote t:=(ppq+1) then, if n3,

    (tn1)={ppq+1if q>1+pn1n1n2if q1+pn1, (4.2)

    if instead n=2 than (tn1) is any real number greater than 1.

    Let p,q satisfy (3.6) and (3.7). It is easy to prove that

    ppq+1<q. (4.3)

    In the following it will be useful to introduce the following notation:

    ν:=1(ppq+1)1p,

    or, more explicitly,

    ν={p1pif q1+pn11qp+1n1if q>1+pn1. (4.4)

    Remark 4.2. Assume 1<pq. Then easy computations give

    ν>0q<pnn1,ν=0q=pnn1. (4.5)

    To get the sharp bound for q, we use a result proved in [43], see also [3,4,30,58]. Here we denote Sσ(x0) the boundary of the ball Bσ(x0) in Rn.

    Lemma 4.3. Let nN, n2. Consider Bσ(x0) ball in Rn and uL1(Bσ(x0)) and s>1. For any 0<ρ<σ<+, define

    I(ρ,σ,u):=inf{Bσ(x0)|u||Dη|sdx:ηC10(Bσ(x0)), 0η1, η=1  in  Bρ(x0)}.

    Then for every δ]0,1],

    I(ρ,σ,v)(σρ)s1+1δ(σρ(Sr(x0)|v|dHn1)δdr)1δ.

    The following result is the Sobolev inequality on spheres.

    Lemma 4.4. Let nN, n3, and γ[1,n1[. Then there exists c depending on n and γ such that for every uW1,p(S1(x0),dHn1)

    (S1(x0)|u|(γn1)dHn1)1(γn1)c(S1(x0)(|Du|γ+|u|γ)dHn1)1γ.

    Lemma 4.5. Let n=2. Then there exists c such that for every uW1,1(S1(x0),dH1) and every r>1,

    (S1(x0)|u|rdH1)1rc(S1(x0)(|Du|+|u|)dH1).

    Proof. By the one-dimensional Sobolev inequality

    uL(S1(x0))cuW1,1(S1(x0)).

    Then, for every r>1,

    (S1(x0)|u|rdHn1)1rcuL(S1(x0))cuW1,1(S1(x0)).

    We conclude this section, by stating a classical result; see, e.g., [41]. that will be useful to prove Theorem 3.2.

    Lemma 4.6. Let α>0 and (Jh) a sequence of real positive numbers, such that

    Jh+1AλhJ1+αh,

    with A>0 and λ>1.

    If J0A1αλ1α2, then JhλhαJ0 and limhJh=0.

    Under the assumptions in Section 3 we have the following Caccioppoli-type inequality.

    Given a measurable function u:ΩR, with Ω open set in Rn, and fixed x0Rn, kR and τ>0, we denote the super-level set of u as follows:

    Ak,τ(x0):={xBτ(x0):u(x)>k};

    usually dropping the dependence on x0. We denote |Ak,τ| its Lebesgue measure.

    Proposition 5.1 (Caccioppoli's inequality). Let uW1,qloc(Ω) be a weak solution to (3.1). If (3.6)–(3.10) hold true, then there exists a constant c>0, such that for any BR0(x0)Ω, 0<ρ<RR0

    Bρ|D(uk)+|pdxC(n,p,q,R0)(Rρ)(ppq+11+ppq+1(ppq+1))××(uk)+ppq+1W1,p(BR)|Ak,R|ppq+1ν+c(uk)+pγ1q1W1,p(BR)|Ak,R|1pγ1p(q1)+c(uk)+pprW1,p(BR)|Ak,R|11pprp+c(uk)+γ2+1W1,p(BR)|Ak,R|1γ2+1p+c(uk)+γ2W1,p(BR)|Ak,R|1γ2p+ckγ2(uk)+W1,p(BR)|Ak,R|11p+c(kpγ1q1+kγ2)|Ak,R|+c(uk)+W1,p(BR)|Ak,R|11s21p+c|Ak,R|1ps1(q1) (5.1)

    with ν as in (4.4) and c is a constant depending on n,p,q,r,R0, the Ls1-norm of b1 and the Ls2-norm of b2 in BR0.

    Proof. Without loss of generality we assume that the functions b1,b2 in (3.3) are a.e. greater than or equal to 1 in Ω. We split the proof into steps.

    Step 1. Consider BR0(x0)Ω, 0<R02ρ<RR01.

    We set

    A(ρ,R):={ηC0(BR(x0)):η=1in Bρ(x0), 0η1}. (5.2)

    For every ηA(ρ,R) and fixed k>1 we define the test function φk as follows

    φk(x):=(u(x)k)+[η(x)]μfor a.e. xBR0(x0),

    with

    μ:=ppq+1 (5.3)

    that is greater than 1 because q>1.

    Notice that φkW1,q0(BR0(x0)), suppφkBR(x0).

    Step 2. Let us consider the super-level sets:

    Ak,R:={xBR(x0):u(x)>k}.

    In this step we prove that

    Ak,ρ|Du|pdxc{Ak,R|Dη|ppq+1(uk)ppq+1dx+Ak,R((uk)pγ1q1+(uk)ppr+(uk)γ2+1+(uk)γ2)dx+cAk,R(kγ2(uk)+b2(uk)+kpγ1q1+kγ2+bpq11)dx} (5.4)

    for some constant c independent of u and η.

    Using φk as a test function in (3.5) we get

    I1:=Ak,Ra(x,u,Du),Duημdx=μAk,Ra(x,u,Du),Dηημ1(uk)dxAk,Rb(x,u,Du)(uk)ημdx=:I2+I3. (5.5)

    Now, we separately consider and estimate Ii, i=1,2,3.

    ESTIMATE OF I3

    Using (3.4) we obtain

    I3ΛAk,Rημ{|Du|r(uk)+|u|γ2(uk)+b2(uk)}dx.

    We estimate the right-hand side using the Young inequality, with exponents pr and ppr, and (3.2). There exists c, depending on λ, Λ, n, p, r, such that

    Λ|Du|r(uk)λ4|Du|p+c(uk)ppr14a(x,u,Du),Du+c(uk)ppra.e. in {|Du|1}. (5.6)

    and, recalling that b21,

    Λ|Du|r(uk)Λ(uk)Λb2(uk)a.e. in {|Du|<1}.

    Therefore,

    I314Ak,R{|Du|1}a(x,u,Du),Duημdx+cAk,Rημ{(uk)ppr+|u|γ2(uk)+b2(uk)}dx. (5.7)

    Collecting (5.5)–(5.7) we get

    34Ak,R{|Du|1}a(x,u,Du),DuημdxI2Ak,R{|Du|1}a(x,u,Du),Duημdx+cAk,Rημ{(uk)ppr+|u|γ2(uk)+b2(uk)}dx.

    Using (3.2) and (3.3) we get

    3λ4Ak,R{|Du|1}|Du|pημdxI2+2ΛAk,R{|Du|1}(|u|γ2+b1)ημdx+cAk,Rημ{(uk)ppr+|u|γ2(uk)+b2(uk)}dx. (5.8)

    ESTIMATE OF I2. For a.e. xAk,R{η0} we have

    μ|a(x,u,Du),Dη|(uk)ημ1μΛ{|Du|q1+|u|γ1+b1}|Dη|(uk)ημ1. (5.9)

    For a.e. x{|Du|1}Ak,R{η0}, by q<p+1 and the Young inequality with exponents pq1 and ppq+1, and noting that μ1=μq1p, we get

    μΛ|Du|q1|Dη|(uk)ημ1λ4|Du|pημ+c(λ,Λ)μppq+1|Dη|ppq+1(uk)ppq+1. (5.10)

    On the other hand we have

    μΛ|Du|q1|Dη|(uk)ημ1μΛ|Dη|(uk)ημ1 (5.11)

    a.e. in {|Du|<1}Ak,R{η0}.

    Therefore,

    I2λ4Ak,R{|Du|1}|Du|pημdx+c(λ,Λ)μppq+1Ak,R{|Du|1}|Dη|ppq+1(uk)ppq+1dx+Ak,R|Dη|(uk)ημ1dx+cAk,R|Dη|ημ1{|u|γ1+b1}(uk)dx.

    By (5.8) and the inequality above, we get

    λ2Ak,R{|Du|1}|Du|pημdxc(λ,Λ,p,q)Ak,R|Dη|ppq+1(uk)ppq+1dx+Ak,R|Dη|ημ1(|u|γ1+b1)(uk)dx+cAk,Rημ((uk)ppr+|u|γ2(uk)+|u|γ2+b2(uk)+b1)dx.

    Taking into account that b11

    Ak,R|Du|pημdx=Ak,R{|Du|1}|Du|pημdx+Ak,R{|Du|<1}|Du|pημdxAk,R{|Du|1}|Du|pημdx+Ak,Rb1ημdx,

    therefore

    Ak,R(|Du|pb1)ημdxAk,R{|Du|1}|Du|pημdx

    and we obtain

    Ak,ρ|Du|pdxcAk,R|Dη|ppq+1(uk)ppq+1dx+Ak,R|Dη|ημ1(|u|γ1+b1)(uk)dx+cAk,Rημ((uk)ppr+|u|γ2(uk)+|u|γ2+b2(uk)+b1)dx. (5.12)

    We have

    Ak,R|Dη|ημ1|u|γ1(uk)dxc(γ1)Ak,R|Dη|ημ1(uk)γ1+1dx
    +c(γ1)Ak,R|Dη|ημ1kγ1(uk)dx.

    By Hölder inequality with exponents pq1 and ppq+1, we get

    Ak,R|Dη|ημ1(uk)γ1+1dx=Ak,R|Dη|(uk)ημ1(uk)γ1dx
    cAk,R|Dη|ppq+1(uk)ppq+1dx+cAk,Rηp(μ1)q1(uk)pγ1q1dx.

    Analogously,

    Ak,R|Dη|ημ1kγ1(uk)dxcAk,R|Dη|ppq+1(uk)ppq+1dx
    +cAk,Rηp(μ1)q1kpγ1q1dx

    and

    Ak,R|Dη|ημ1b1(uk)dxcAk,R|Dη|ppq+1(uk)ppq+1dx
    +cAk,Rηp(μ1)q1bpq11dx,

    obtaining

    Ak,ρ|Du|pdxc{Ak,R|Dη|ppq+1(uk)ppq+1dx+Ak,R((uk)pγ1q1+kpγ1q1+bpq11)dx+cAk,R((uk)ppr+|u|γ2(uk)+|u|γ2+b2(uk)+b1)dx.}.

    Therefore,

    Ak,ρ|Du|pdxc{Ak,R|Dη|ppq+1(uk)ppq+1dx+Ak,R((uk)pγ1q1+(uk)ppr+(uk)γ2+1+(uk)γ2)dx+cAk,R(kγ2(uk)+kγ2+b2(uk)+b1+kpγ1q1+bpq11)dx.}.

    Since b11 and q<p+1, then

    b1+bpq112bpq11,

    and we get (5.4).

    Step 3. In this step we prove that

    Bρ|D(uk)+|pdxC(n,p,q,R0)(Rρ)(ppq+11+ppq+1(ppq+1))××(uk)+ppq+1W1,p(BR(x0))|Ak,R|ppq+1ν+cAk,R((uk)pγ1q1+(uk)ppr+(uk)γ2+1+(uk)γ2)dx+cAk,R(kγ2(uk)+b2(uk)+kpγ1q1+kγ2+bpq11)dx. (5.13)

    We obtain this estimate starting by (5.4).

    Consider τ(ρ,R) and define the function

    S1(0)yw(y):=(uk)+(x0+τy)

    where

    S1(0):={yRn:|y|=1}.

    This function w is in W1,(ppq+1)(S1,dHn1), with

    1(ppq+1)=min{1ppq+1+1n1,1}. (5.14)

    Let us consider the case

    q>1+pn1.

    By (4.1) in Remark 4.1, we get

    1(ppq+1)=1ppq+1+1n1. (5.15)

    By (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we get

    (S1|w|ppq+1dHn1)pq+1pc(n,p,q)(S1(|Dw|(ppq+1)+|w|(ppq+1))dHn1)1/(ppq+1). (5.16)

    When

    q1+pn1,

    we distinguish among two cases: n3 and n=2. If n3, by using Hölder's inequality, we get

    (S1|w|ppq+1dHn1)pq+1pc(n,p,q)(S1|w|n1n2dHn1)n2n1,

    by (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we obtain the inequality (5.16).

    If n=2, then (ppq+1)=1, then we obtain the inequality (5.16) by applying Lemma 4.5 with r=ppq+1.

    Let A(ρ,R) be as in (5.2). We apply Lemma 4.3, with

    BR(x0)yv(y):=(uk)ppq+1+(y),

    that is a function in L1(BR(x0)). Using (5.16) and recalling that R02ρ<RR0, reasoning as in [30], we get

    infA(ρ,R)BR(x0)|Dη|ppq+1(uk)ppq+1+dxC(n,p,q,R0)(Rρ)(ppq+11+ppq+1(ppq+1))××(RρSτ(0)(|D(uk)+(x0+y)|(ppq+1)+|(uk)+(x0+y)|(ppq+1))dHn1(y)dτ)ppq+1/(ppq+1). (5.17)

    By coarea formula, inequality (5.17) implies

    infA(ρ,R)BR(x0)|Dη|ppq+1(uk)ppq+1+dxC(n,p,q,R0)(Rρ)(ppq+11+ppq+1(ppq+1))××(uk)+ppq+1W1,(ppq+1)(BR(x0)Bρ(x0))

    and, taking into account (3.7), Remark 4.1 and (4.5)

    (ppq+1)<p1(ppq+1)>1pν>0qp<1+1n1,

    by Hölder's inequality we get

    infA(ρ,R)BR(x0)|Dη|ppq+1(uk)ppq+1+dxC(n,p,q,R0)(Rρ)(ppq+11+ppq+1(ppq+1))××(uk)+ppq+1W1,p(BR(x0))|Ak,R|ppq+1ν (5.18)

    By (5.4) we get

    Ak,ρ|Du|pdxC(n,p,q,R0)(Rρ)(ppq+11+ppq+1(ppq+1))××(uk)+ppq+1W1,p(BR(x0))|Ak,R|ppq+1ν+cAk,R((uk)pγ1q1+(uk)ppr+(uk)γ2+1+(uk)γ2)dx+cAk,R(kγ2(uk)+b2(uk)+kpγ1q1+kγ2+bpq11)dx.

    Since

    Bρ|D(uk)+|pdx=Ak,ρ|D(uk)+|pdx=Ak,ρ|Du|pdx

    we get (5.13).

    Step 4. In this step we estimate the integrals at the right hand side of (5.13).

    Consider

    J1:=Ak,R((uk)pγ1q1+(uk)ppr+(uk)γ2+1+(uk)γ2)dx.

    ESTIMATE OF J1.

    By assumptions (3.8) and (3.9),

    max{pγ1q1,γ2+1,ppr}<p.

    Therefore, by using Hölder inequality with exponent p(q1)pγ1 we get

    Ak,R(uk)pγ1q1dx(Ak,R(uk)pdx)pγ1p(q1)|Ak,R|1pγ1p(q1);

    Hölder inequality with exponent pprp implies

    Ak,R(uk)pprdx(Ak,R(uk)pdx)1pprp|Ak,R|11pprp.

    Moreover, by using Hölder inequality with exponent pγ2+1 we get

    Ak,R(uk)γ2+1dx(Ak,R(uk)pdx)γ2+1p|Ak,R|1γ2+1p;

    by using Hölder inequality with exponent pγ2 we get

    Ak,R(uk)γ2dx(Ak,R(uk)pdx)γ2p|Ak,R|1γ2p.

    Therefore, by using the Sobolev embedding theorem

    J1(uk)+pγ1q1W1,p(BR)|Ak,R|1pγ1p(q1)+(uk)+pprW1,p(BR)|Ak,R|11pprp+(uk)+γ2+1W1,p(BR)|Ak,R|1γ2+1p+(uk)+γ2W1,p(BR)|Ak,R|1γ2p.

    Let us consider now the following integral in (5.13):

    J2:=Ak,R(kγ2(uk)+b2(uk)+kpγ1q1+kγ2+bpq11)dx.

    Trivially,

    Ak,Rkγ2(uk)dxkγ2(uk)+1pLp(Ak,R)|Ak,R|11pkγ2(uk)+W1,p(Ak,R)|Ak,R|11p.

    By assumption b2Ls2, s2>np=ppp. Since ppp>pp1, then s2s21<p. Therefore, by Hölder inequality

    Ak,Rb2(uk)dxb2Ls2(Ak,R)(uk)+Ls2s21b2Ls2(BR)(uk)+Lp(Ak,R)|Ak,R|11s21p,

    which implies

    Ak,Rb2(uk)dxb2Ls2(BR)(uk)+W1,p(BR)|Ak,R|11s21p.

    Now, b1Ls1 with s1>pq1; by using Hölder inequality with exponent s1(q1)p we get

    Ak,Rbpq11dx(Ak,Rbs11dx)ps1(q1)|Ak,R|1ps1(q1).

    We obtain

    J2kγ2(uk)+W1,p((BR))|Ak,R|11p+(kpγ1q1+kγ2)|Ak,R|+b2Ls2(BR)(uk)+W1,p(BR)|Ak,R|11s21p+b1pq1Ls1(BR)|Ak,R|1ps1(q1).

    Step 5. By Steps 3, 4 we get

    Br|D(uk)+|pdxC(n,p,q,R0)(Rρ)(ppq+11+ppq+1(ppq+1))××(uk)+ppq+1W1,p(BR)|Ak,R|ppq+1ν+c(uk)+pγ1q1W1,p(BR)|Ak,R|1pγ1p(q1)+c(uk)+pprW1,p(BR)|Ak,R|11pprp+c(uk)+γ2+1W1,p(BR)|Ak,R|1γ2+1p+c(uk)+γ2W1,p(BR)|Ak,R|1γ2p+ckγ2(uk)+W1,p(BR)|Ak,R|11p+c(kpγ1q1+kγ2)|Ak,R|+cb2Ls2(BR)(uk)+W1,p(BR)|Ak,R|11s21p+cb1pq1Ls1(BR)|Ak,R|1ps1(q1)

    and the inequality (5.1) follows.

    Let , , be weak solution to (3.1). Consider an open set.

    I case . Let .

    For every

    (6.1)

    In particular, chosen such that

    we get

    (6.2)

    II case . By a well known result by Giaquinta and Giusti [40], the gradient of the weak solution satisfies a higher integrability property: its gradient is in , for some sufficiently small. Moreover, ; because , we can repeat the above argument with replaced by so obtaining (6.1). depends on the norm . Again, by the Giaquinta and Giusti result, the norm can be estimated in terms of the for .

    Finally, we can summarize: in both cases, either if or if , we can choose such that (6.2) holds with depending on the norm . We also assume such that , .

    Define the decreasing sequences

    Fixed a positive constant , to be chosen later, define the increasing sequence of positive real numbers

    Define the decreasing sequence ,

    Notice that

    Moreover, by (6.2),

    Let us introduce the following notation:

    (6.3)
    (6.4)

    and

    (6.5)

    where is defined in (4.4).

    Proposition 6.1 (Estimate of ). Let be a weak solution to (3.1). Assume (3.2)–(3.4) with the exponents satisfying the inequalities listed in Section 3.1. Then for every

    (6.6)

    where is a constant depending on , the -norm of and the -norm of in .

    We precede the proof with the following remark.

    Remark 6.2. We remark that, by assumptions (3.6)–(3.10), then and . As far as these inequalities are concerned, we remark that

    that is satisfied, because

    that is the first assumption in (3.10); this assumption also implies

    that is equivalent to

    By the second assumption in (3.10),

    Proof of Proposition 6.1. By (5.1), used with , , , we have

    (6.7)

    Let us write the estimate above as

    (6.8)

    To estimate the sum at the right-hand side it is useful to remark that, for all ,

    (6.9)

    and

    Since

    by the Sobolev inequality we get

    that, together with (6.9), gives

    (6.10)

    Moreover,

    (6.11)

    Inequalities (6.10) and (6.11) imply that

    therefore, by (6.9),

    (6.12)

    This estimate, together with (6.10), implies:

    (6.13)

    and, analogously,

    (6.14)
    (6.15)
    (6.16)
    (6.17)
    (6.18)

    Moreover, taking into account that

    (6.19)
    (6.20)

    Let us now estimate .

    Inequalities (6.10) and (6.11) imply

    that gives

    Taking into account that for every

    we conclude that

    (6.21)

    Collecting (6.13)–(6.21), by (6.8) we get

    (6.22)

    Let us now add to both sides of (6.22) the integral .

    By Hölder inequality

    Since

    the Sobolev embedding theorem gives

    (6.23)

    Taking into account (6.10), we obtain

    therefore, the inequality (6.23) implies

    (6.24)

    Inequalities (6.22) and (6.24) give

    (6.25)

    where is a constant depending on , the -norm of and the -norm of in .

    By taking in account the notation in (6.3)–(6.5), we get, by (6.25), the inequality (6.6).

    We are now ready to prove our regularity result.

    Proof of Theorem 3.2. By Proposition 6.1, for every ,

    where is a constant depending on , the -norm of and the -norm of in and for every . Thus, the following inequality holds:

    with

    where , and are defined in (6.4), (6.3), (6.5). We recall that , see Remark 6.2.

    To apply Lemma 4.6, we need

    (6.26)

    Since

    if we choose satisfying

    (6.27)

    we get and we conclude that

    To prove that is locally bounded from below, we proceed as follows. The function is a weak solution to

    where

    Notice that, by (3.2)–(3.4) the following properties hold:

    ellipticity condition at infinity:

    for a.e. and for every ,

    growth condition:

    for a.e. and every and

    growth condition for the right hand side :

    To prove the analogue of Proposition 5.1 we now consider the test function where is a cut-off function. Let us consider the sub-level sets:

    Then we obtain, in place of (5.5),

    The proof goes on with no significant changes with respect the previous case, arriving to the conclusion that there exists such that we obtain that , and

    Collecting the estimates from below and from above for , we conclude.

    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).

    The authors declare no conflict of interest.

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