arXiv subjects
Publications and source records attributed to Dimitri Mugnai.
We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.
We study the null controllability for a degenerate/singular wave equation with drift in non divergence form. In particular, considering a control localized on the non degenerate boundary point, we provide some conditions for the boundary controllability via energy methods and boundary observability.
We prove the existence and multiplicity of weak solutions for a mixed local-nonlocal problem at resonance. In particular, we consider a not necessarily positive operator which appears in models describing the propagation of flames. A careful adaptation of well known variational methods is required to deal with the possible existence of negative eigenvalues.
We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.
We prove the existence of a weak solution for boundary value problems driven by a mixed local--nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive definite.
We study a nonlinear, nonlocal Dirichlet problem driven by the fractional p-Laplacian, involving a (p-1)-sublinear reaction. By means of a weak comparison principle we prove uniqueness of the solution. Also, comparing the problem to 'asymptotic' weighted eigenvalue problems for the same operator, we prove a necessary and sufficient condition for the existence of a solution. Our work extends classical results due to Brezis-Oswald and Diaz-Saa to the quasilinear nonlocal framework.
We consider a degenerate wave equation with drift in presence of a leading operator which is not in divergence form. We provide some conditions for the boundary controllability of the associated Cauchy problem.
In this paper we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e., $$\mathcal{L}_{p,s} = -\Delta_p + (-\Delta)^s_p.$$ Our main result is resemblant to the celebrated work by Brezis-Oswald [10]. In addition, we prove a regularity result of independent interest.
We show the existence of nontrivial solutions for a class of highly quasilinear problems in which the governing operators depend on the unknown function. By using a suitable variational setting and a weak version of the Cerami-Palais-Smale condition, we establish the desired result without assuming that the nonlinear source satisfies the Ambrosetti-Rabinowitz condition.
We consider nonlinear problems governed by the fractional $p-$Laplacian in presence of nonlocal Neumann boundary conditions. We face two problems. First: the $p-$superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.
We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.
In this paper, first we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sublevels of the associated functional, which permits to apply a critical point theorem of mixed nature due to Marino and Saccon.
We consider a class of pseudo-relativistic Hartree equations in presence of general nonlinearities not satisfying the Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory, we show the existence of two non trivial signed solutions, one positive and one negative.
We establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non smooth coefficients, thus preventing the use of standard calculations in this framework.
The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-Δ)^s u-λu=f(x,u) & {\mbox{ in }} Ω\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus Ω\,, \end{array} \right. $$ where $s\in (0,1)$ is fixed, $(-Δ)^s$ is the fractional Laplace operator, $λ$ is a real parameter, $Ω\subset \mathbb{R}^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-Δ)^s$. At this purpose we employ a variational theorem of mixed type (one of the so-called $\nabla$-theorems).
We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.
We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.
We study the existence of cylindrically symmetric electro-magneto-static solitary waves for a system of a nonlinear Klein-Gordon equation coupled with Maxwell's equations in presence of a positive mass and of a nonnegative nonlinear potential. Nonexistence results are provided as well.