arXiv subjects

Luigi Muglia

Publications and source records attributed to Luigi Muglia.

9 recordsLinked to original sources

arXiv2026

Monotonicity of non-negative solutions of quasilinear elliptic equations in a cylindrical domain

We consider weak solutions to $p$-Laplace equations in cylindrical domains under mixed homogeneous Dirichlet-Neumann boundary conditions. We assume that the right-hand side is positive and locally Lipschitz continuous and we prove that any positive solution is monotone increasing in the $x_N$ direction for any $p>1$. As an application we prove that solutions to Allen-Cahn type equations are one-dimensional as well as a Liouville type result for Lane-Emden type equations.

math.AP↗

arXiv2026

Classification of singular solutions to the subcritical Lane-Emden equation in $\mathbb{R}^N \setminus \{0\}$

We provide a classification result of positive solutions with a non-removable singularity at the origin to the subcritical Lane-Emden equation $-Δ_p u = u^q$ in $\mathbb{R}^N \setminus \{0\}$, where $p\in(1,N)$, $q\in (p-1,p^*-1)$ and $u^{q-p+1} \in L^{\frac{N}{p}}(\mathbb{R}^N \setminus B_R)$.

math.AP↗

arXiv2025

The Classification of all weak solutions to $-Δu={u^{-γ}}$ in the half-space

We provide the classification of all the positive solutions to $-Δu=\frac{1}{u^γ}$ in the half space, under minimal assumption.

math.AP↗

arXiv2025

Monotonicity results in half spaces for quasilinear elliptic equations involving a singular term

We consider positive solutions to $\displaystyle -Δ_p u=\frac{1}{u^γ}+f(u)$ under zero Dirichlet condition in the half space. Exploiting a prio-ri estimates and the moving plane technique, we prove that any solution is monotone increasing in the direction orthogonal to the boundary.

math.AP↗

arXiv2025

Second order regularity for solutions to anisotropic degenerate elliptic equations

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.

math.AP↗

arXiv2024

Regularity and symmetry results for the vectorial p-Laplacian

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.

math.AP↗

arXiv2023

Optimal second order boundary regularity for solutions to $p$-Laplace equations

Solutions to $p$-Laplace equations are not, in general, of class $C^2$. The study of Sobolev regularity of the second derivatives is, therefore, a crucial issue. An important contribution by Cianchi and Maz'ya shows that, if the source term is in $L^2$, then the field $|\nabla u|^{p-2}\nabla u$ is in $W^{1,2}$. The $L^2$-regularity of the source term is also a necessary condition. Here, under suitable assumptions, we obtain sharp second order estimates, thus proving the optimal regularity of the vector field $|\nabla u|^{p-2}\nabla u$, up to the boundary.

math.AP↗

arXiv2023

Classification of solutions to $Δu = u^{-γ}$ in the half-space

We provide a classification result for positive solutions to $Δu = u^{-γ}$ in the half space, under zero Dirichlet boundary condition.

math.AP↗

arXiv2022

Solutions to nonlocal evolution equations governed by non-autonomous forms and demicontinuous nonlinearities

We deal with the existence of solutions having L2 regularity for a class of non autonomous evolution equations. Associated with the equation, a general non local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray-Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.

math.AP↗