arXiv subjects
Publications and source records attributed to L. Del Pezzo.
In this paper we consider the blow-up problem for a mixed local-nonlocal diffusion operator, \[ u_t=aΔu -b(-Δ)^s u+u^p. \] We show that the Fujita exponent is given by the nonlocal part, $p_F=1+2s/N$. We also determinate, in some cases, the blow-up rate.
In this paper we analyze the semi-linear fractional Laplace equation $$(-Δ)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in \mathbb{R}^N:\ x_N>0\}$ stands for the half-space and $f$ is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if $u$ is a bounded solution with $ρ:=\sup_{\mathbb{R}^N}u$ verifying $f(ρ)=0$, then $u$ is necessarily one-dimensional.
In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-Δ)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-Δ)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the functions $h$ and $f$ are nondecreasing. The main feature is that the function $h$ changes sign in $\mathbb{R}$, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
In this paper we analyze an eigenvalue problem related to the nonlocal $p-$laplace operator plus a potential. After reviewing some elementary properties of the first eigenvalue of these operators (existence, positivity of associated eigenfunctions, simplicity and isolation) we investigate the dependance of the first eigenvalue on the potential function and establish the existence of some {\em optimal} potentials in some admissible classes.
In this paper we consider classical solutions $u$ of the semilinear fractional problem $(-Δ)^s u = f(u)$ in $\mathbb{R}^N_+$ with $u=0$ in $\mathbb{R}^N \setminus \mathbb{R}^N_+$, where $(-Δ)^s$, $0 0\}$ is the half-space and $f\in C^1$ is a given function. With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\mathbb{R}^N_+$ and verify $$ \frac{\partial u}{\partial x_N}>0 \quad \hbox{in } \mathbb{R}^N_+. $$ This is in contrast with previously known results for the local case $s=1$, where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when $f(0)<0$.
In this paper we show existence of solutions for some elliptic problems with nonlocal diffusion by means of nonvariational tools. Our proof is based on the use of topological degree, which requires a priori bounds for the solutions. We obtain the a priori bounds by adapting the classical scaling method of Gidas and Spruck. We also deal with problems involving gradient terms.
In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional $\J(u)=\int_{\partialΩ} f(x) u \rd \H^{N-1}$ over some admissible class of loads $f$ where $u$ is the (unique) solution to the problem $-Δ_p u + |u|^{p-2}u = 0$ in $Ω$ with $|\nabla u|^{p-2}u_ν= f$ on $\partial Ω$.