On the comparison principle for unbounded solutions of elliptic equations with first order terms

We prove a comparison principle for unbounded weak sub/super solutions of the equation λu − div(A(x)Du) = H(x, Du) in Ω where A(x) is a bounded coercive matrix with measurable ingredients, λ ≥ 0 and ξ → H(x, ξ) has a super linear growth and is convex at infinity. We improve earlier results where the convexity of H(x, ·) was required to hold globally

Equazioni ellittiche con termine a crescita naturale nel gradiente e dati misura

Trabajo fin de carrera (Tesi di Laurea)Matematica. Anno Accademico 2001-200

Gradient estimates for quasilinear elliptic Neumann problems with unbounded first-order terms

This paper studies global a priori gradient estimates for divergence-type equations patterned over the $p$-Laplacian with first-order terms having polynomial growth with respect to the gradient, under suitable integrability assumptions on the source term of the equation. The results apply to elliptic problems with unbounded data in Lebesgue spaces complemented with Neumann boundary conditions posed on convex domains of the Euclidean space

The best approximation of a given function in L2L^2-norm by Lipschitz functions with gradient constraint

The starting point of this paper is the study of the asymptotic behavior, as $p\to\infty$, of the following minimization problem $$\min\left\{\frac1{p}\int|\nabla v|^{p}+\frac12\int(v-f)^2 \,, \quad \ v\in W^{1,p} (\Omega)\right\}.$$ We show that the limit problem provides the best approximation, in the $L^2$-norm, of the datum $f$ among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover such approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the $(N-1)$-Hausdorff measure of the jump set of the function

Principal Eigenvalue of Mixed Problem for the Fractional Laplacian: Moving the Boundary Conditions

We analyze the behavior of the eigenvalues of the following non local mixed problem \left\{ \begin{array}{rcll} (-\Delta)^{s} u &=& \lambda_1(D) \ u &\inn\Omega,\\ u&=&0&\inn D,\\ \mathcal{N}_{s}u&=&0&\inn N. \end{array}\right Our goal is to construct different sequences of problems by modifying the configuration of the sets $D$ and $N$, and to provide sufficient and necessary conditions on the size and the location of these sets in order to obtain sequences of eigenvalues that in the limit recover the eigenvalues of the Dirichlet or Neumann problem. We will see that the non locality plays a crucial role here, since the sets $D$ and $N$ can have infinite measure, a phenomenon that does not appear in the local case (see for example \cite{D,D2,CP})