We extend several characterizations of the $L^p$-solvability of the homogeneous Dirichlet problem to (possibly degenerate) elliptic operators $L=-\textrm{div}(wA\nabla)$ defined on a large class of open sets $\Omega$ in $\mathbb{R}^n$. This framework encompasses not only uniformly elliptic operators in Lipschitz domains, but also Caffarelli-Sylvestre-type operators, and boundaries $\partial\Omega$ that are not $(n-1)$-dimensional, for instance. We prove that, for $p\in (1,\infty)$, the $L^p$-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem $Lu=wf-\textrm{div}(wF)$, \emph{i.e.} to the existence of a solution $u$ satisfying a $L^p$ non-tangential estimate whenever $f$ and $F$ belong to suitable weighted $L^p$ tent spaces. Furthermore, it is also equivalent to the solvability of the Poisson-Regularity problem for data in appropriate weighted $L^{p'}$ tent spaces, where estimates are obtained for $\nabla u$ rather than $u$.
We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$ of convex points $x \in \partial\Omega$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(\Gamma_0)>\frac{2}{3}\mathcal{H}^{n-1}(\partial\Omega)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\Omega$ and attains the prescribed boundary data at least on $\Gamma_0$ in the classical sense.
We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $\lambda_1(D)\ge N-1$, where $\lambda_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-\Delta_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.
Carlo Alberto Antonini, F. Pacella, Camilla Chiara Polvara et al.· 0 citations
In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally H\"older continuous. Suppose that $Du\in C^{0,\alpha}_{\rm loc}$ with $\alpha\le 1/(p-1)$, and let $\Phi$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $\Phi(Du)\in C^k_{\rm loc}$ whenever $m>k/\alpha$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du|^{1/\alpha}$, Schauder estimates for a normalized uniformly elliptic equation, and an extension lemma across the critical set. We also obtain corresponding results for autonomous anisotropic equations and for elliptic and parabolic $p$-Laplace systems, under the appropriate H\"older assumption on the gradient. Finally, the same argument gives $C^k$ regularity criteria for high powers of nonnegative solutions to the porous medium equation.
We study the uniform $L^\infty(\Omega)$ a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in a bounded, convex, $C^{1,1}$ domain $\Omega \subset \mathbb{R}^N$ with homogeneous exterior condition $u\equiv 0$ in $\mathbb{R}^N\setminus\Omega$. We consider slightly superlinear nonlinearities of the form $f(t) = t^q L(t)$, where $1 \le q \le \frac{N+2s}{N-2s}$ and $L$ is a slowly varying function. Although uniform estimates are well-established in the strictly subcritical regime $q<\frac{N+2s}{N-2s}$, the slightly subcritical case, $q = \frac{N+2s}{N-2s}$, is highly challenging due to the potential formation of bubbling profiles. In this work, we isolate a structural condition on the slowly varying perturbation, namely $$ \lim_{t \to \infty} \frac{t \, |L'(t)|}{L^{\frac{N}{2s}}(t)} = \infty, $$ which acts as an asymptotic barrier that prevents mass concentration. Under this assumption, we establish global uniform $L^\infty(\Omega)$ bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.
Juan-Carlos Felipe-Navarro, Rosa Pardo· 0 citations
We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation $\Delta u=e^u$, the real equation $\sigma_2(D^2u)=e^{2u}$, and its complex counterpart $\sigma_2(u_{i\bar j})=e^{2u}$. In each case $u<0$ in the domain and $u=0$ on the boundary. We prove that \[ w=-\operatorname{arcosh}(e^{-u/2}) \] is strictly convex in the underlying real variables. The argument combines domain deformation, constant-rank theory, inverse-convexity estimates, radial ball models, boundary strict convexity, and local $C^2$ stability.
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.
Rubén Fiñana, A. Molino· Nonlinear Analysis· 0 citations
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