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Global integral gradient bounds for quasilinear equations below or near the natural exponent. (English) Zbl 1327.35384

Let \(\Omega\subset\mathbb{R}^n\), \(n\geq 2\), be a bounded domain and let \(\mu\) be a finite measure on \(\Omega\). The author studies the maximal global regularity for gradients of weak solutions to the problem
\[ -\text{div}\mathcal{A}(x,\nabla u)=\mu \text{ in } \Omega, \quad u=0 \text{ on } \partial \Omega,\tag{1} \]
where \(\mathcal{A}:\mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}^n\) is a Carathéodory function satisfying the condition that for some \(p\in [2-\frac{1}{n},n]\) and \(\alpha,\beta>0\) there holds
\[ |\mathcal{A}(x,\xi)|\leq \beta |\xi|^{p-1}\text{ and } \langle \mathcal{A}(x,\xi)-\mathcal{A}(x,\eta),\xi-\eta\rangle\geq \alpha (|\xi|^2+|\eta|^2)^{\frac{p}{2}-1}|\xi-\eta|^2\tag{2} \]
for every \((\xi,\eta)\in \mathbb{R}^n\times\mathbb{R}^n\setminus \{(0,0)\}\) and almost every \(x\in \mathbb{R}^n\).
The author assumes the following capacitary density condition on the domain \(\Omega\): There exist \(c_0,r_0\in (0,\infty)\) such that \[ \text{cap}_p(\overline{B_t(x)}\cap (\mathbb{R}^n\setminus \Omega),B_{2t}(x))\geq \text{cap}_p(\overline{B_t(x)},B_{2t}(x)) \]
for all \(t\in (0,r_0]\) and all \(x\in \mathbb{R}^n\setminus \Omega\). Here, the \(p\)-capacity \(\text{cap}_p\) is defined by \[ \text{cap}_p(K,B_{2t}(x))=\inf\left\{\int_{B_{2t}(x)}|\nabla \varphi|^pdx: \varphi\in C_0^\infty(B_{2t}(x)),\quad \varphi\geq \chi_K\right\} \]
for all compact set \(K\subset B_{2t}(x)\), where \(\chi_A\) is the characteristic function of \(A\) for all \(A\subset \mathbb{R}^n\). Under this condition on \(\Omega\), which is satisfied, for instance, when \(\Omega\) has a Lipschitz boundary, and under condition \((2)\) on \(\mathcal{A}\), the author establishes the following result: There exists a positive constant \(\varepsilon>0\) depending only on \(n,p,\alpha,\beta\) and \(c_0\) such that for any \(q\in(0,p+\varepsilon)\), any \(t\in (0,\infty]\), and any solution \(u\) to problem \((1)\), the following gradient estimate holds: \[ \|\nabla u\|_{L^{q,t}(\Omega)}\leq C\|\mathcal{M}_1(\chi_\Omega|\mu|)^{\frac{1}{p-1}}\|_{L^{q,t}(\mathbb{R}^n)}.\tag{3} \]
Here, \(C\) is a positive constant which depends only on \(n,p,q,t,c_0\) and \(\text{diam}(\Omega)/r_0\), and
\[ \mathcal{M}_1(\nu)(x):=\sup_{r>0}\frac{r\nu(B_r(x))}{|B_r(x)|}, \quad x\in \mathbb{R}^n, \]
is the fractional maximal function defined for each nonnegative locally finite measure \(\nu\) in \(\mathbb{R}^n\). This result improves a local version of inequality \((3)\) given in the article [G. Mingione, Math. Ann. 346, No. 3, 571–627 (2010; Zbl 1193.35077)] in the case \(p\in [2,n]\). As a corollary to inequality \((3)\), for \(\mu=f\in L^{\gamma,t}(\Omega)\) with \(\gamma \in (1,n(p+\varepsilon)/n(p-1)+p+\varepsilon)\), using the boundedness property of the fractional maximal function on Lorentz spaces, one obtains the gradient estimate
\[ \||\nabla u|^{p-1}\|_{L^{n\gamma/(n-\gamma),t}(\Omega)}\leq C\|f\|_{L^{\gamma,t}(\Omega)}. \]
A further application of the main result concerns a Calderón-Zygmund type estimate below the natural exponent \(p\) for \(\mathcal{A}\)-superharmonic solutions to the equation
\[ \text{div} \mathcal{A}(x,\nabla u)=\text{div} F, \quad \text{in } \mathcal{D}'(\mathbb{R}^n).\tag{4} \]
More precisely, the author proves that for any \(q\in (\max\{1,p-1\},p)\), any \(t\in (0,\infty]\), and any vector field \(F\in L^{q/(p-1),t/(p-1)}(\mathbb{R}^n,\mathbb{R}^n)\), the gradient estimate
\[ \|\nabla u\|_{L^{q,t}(\mathbb{R}^n)}\leq C\|\nabla u\|_{L^1(\mathbb{R}^n)}+C\|F\|^{1/(p-1)}_{L^{q/(p-1),t/(p-1)}(\mathbb{R}^n)}\tag{5} \]
holds for any entire \(\mathcal{A}\)-superharmonic solution \(u\) of \((4)\) such that \(\nabla u \in L^1(\mathbb{R}^n,\mathbb{R}^n)\). Here, \(C\) is a positive constant depending only on \(n\), \(p\), \(q\), \(t\), \(\alpha\), and \(\beta\). This gives a partial answer to a question proposed in the article [T. Iwaniec, Stud. Math. 75, 293–312 (1983; Zbl 0552.35034)], where inequality (5) was conjectured to hold for all distributional solutions to equation (4).
Finally, the author considers equation (4) without the assumption \(q>1\), i.e., with \(q\in (p-1,p)\). In this case, the author, under the further assumption that \(-\mathrm{div}F\geq 0\) in \(\mathcal{D}'(\mathbb{R}^n)\), proves that there exists an entire nonnegative \(\mathcal{A}\)-superharmonic solution \(u\) of (4) such that
\[ \|u\|_{L^{nq/(n-q),t}(\mathbb{R}^n)}+\|\nabla u\|_{L^{q,t}(\mathbb{R}^n)}\leq C\|F\|^{1/(p-1)}_{L^{q/(p-1),t/(p-1)}(\mathbb{R}^n)}. \]
The preliminary results to the proof of the main result concern certain interior and boundary comparison estimates for solutions to the equation div\((\mathcal{A}(x,\nabla w))=0\) on suitable domains.

MSC:

35R06 PDEs with measure
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J62 Quasilinear elliptic equations
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