arXiv subjects
Publications and source records attributed to Domenico Vuono.
We prove local second-order regularity near the critical set for widely degenerate elliptic systems, and show that this degenerate region has zero Lebesgue measure provided the source term is non-vanishing.
We provide a classification result of positive solutions with a non-removable singularity at the origin to the subcritical Lane-Emden equation $-\Delta_p u = u^q$ in $\mathbb{R}^N \setminus \{0\}$, where $p\in(1,N)$, $q\in (p-1,p^*-1)$ and $u^{q-p+1} \in L^{\frac{N}{p}}(\mathbb{R}^N \setminus B_R)$.
We study higher regularity for weak solutions of the $p$-Laplace equation $-Δ_p u = f$ in a domain $Ω\subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and Hölder regularity conditions, we prove that the nonlinear quantity $|\nabla u|^{m-2}\nabla u$ belongs to $W^{m-1,q}_{loc}(Ω)$, and that $|\nabla u|^{m-2} D^2u$ belongs to $W^{m-2,q}_{loc}(Ω)$, for any $q\ge 2$. Furthermore, we obtain uniform $L^\infty$ bounds for the weighted $(m-1)$-th derivatives of $|\nabla u|^{m-2}\nabla u$ and the weighted $(m-2)$-th derivatives of $|\nabla u|^{m-2} D^2u$, providing quantitative control even near critical points of $\nabla u$.
We consider the equation $- Δ_{p} u = f(x)$ in $Ω,$ where $Δ_{p}$ is the $p$-Laplace operator. We provide $L^{\infty}$-type estimates for the second derivatives of solutions when $p$ approaches to $2$.
We consider classical solutions to $-Δu = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided that it is directionally bounded on finite strips. As a corollary, we deduce a new Liouville-type theorem for the Lane-Emden equation.
We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the domain is required. As a consequence, with the additional assumption that the source term has a sign, we obtain integrability properties of the inverse of the gradient of the solution. Assuming convexity of the domain, no boundary regularity is required.
We consider solutions to degenerate anisotropic elliptic equations in order to study their regularity. In particular we establish second-order estimates and enclose regularity results for the stress field. All our results are new even in the euclidean case.
We obtain optimal regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x)\,\, \mbox{ in $Ω$}\,.$$ More precisely we address the issue of global second order estimates for the stress field.
We consider weak solutions of the equation $$-Δ_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } Ω,$$ where $H$ is in some cases called Finsler norm, $Ω$ is a domain of $\mathbb R^N$, $p>1$, $q\ge \max\{p-1,1\}$, and $a(\cdot,u)$, $f(\cdot,u)$ are functions satisfying suitable assumptions. We exploit the Moser iteration technique to prove a Harnack type comparison inequality for solutions of the equation and a Harnack type inequality for solutions of the linearized operator. As a consequence, we deduce a Strong Comparison Principle for solutions of the equation and a strong Maximum Principle for solutions of the linearized operator.
We consider solutions to $$ - Δ_{p} u = f(x) \quad \text{in } Ω\, ,$$ when $p$ approaches the semilinear limiting case $p=2$ and we get third order estimates. As a consequence we deduce improved regularity properties of the stress field.
We obtain some regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $Ω$}\,.$$ More precisely we address the issue of second order estimates for the stress field. As a consequence of our regularity results we deduce a weighted Sobolev inequality that leads to weak comparison principles. As a corollary we run over the moving plane technique to deduce symmetry and monotonicity results for the solutions, under suitable assumptions.
The aim of this paper is to deal with the anisotropic doubly critical equation $$-Δ_p^H u - \fracγ{[H^\circ(x)]^p} u^{p-1} = u^{p^*-1} \qquad \text{in } \R^N,$$ where $H$ is in some cases called Finsler norm, $H^\circ$ is the dual norm, $1<p<N$, $0 \leq γ< \left((N-p)/p\right)^p$ and $p^*=Np/(N-p)$. In particular, we provide a complete asymptotic analysis of $u \in \mathcal{D}^{1,p}(\R^N)$ near the origin and at infinity, showing that this solution has the same features of its euclidean counterpart. Some of the techniques used in the proofs are new even in the Euclidean framework.
We prove some Liouville-type theorems for stable solutions (and solutions stable outside a compact set) of quasilinear anisotropic elliptic equations. Our results cover the particular case of the pure Finsler p-Laplacian.
In this paper we introduce and study a new Kelvin-type transform in the anisotropic setting. In particular, we deal with semilinear and quasilinear anisotropic elliptic problems in the entire space.