arXiv subjects

J. F. Rodrigues

Publications and source records attributed to J. F. Rodrigues.

5 recordsLinked to original sources

arXiv2022

On a class of nonlocal problems with fractional gradient constraint

We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type $|D^σu|\leq g$, involving the distributional fractional Riesz gradient $D^σ$, $0<σ<1$, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems. We also prove their convergence as $σ\nearrow1$ towards their local counterparts with the gradient constraint $|D u|\leq g$.

math.AP↗

arXiv2014

On the Regularity of the Free Boundary for Quasilinear Obstacle Problems

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth $p(x)>1$, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for $p(x)$-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of $C^1$ hypersurfaces: i) by extending directly the finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary to the case of heterogeneous $p$-Laplacian type operators with constant $p$; $1 1$.

math.AP↗

arXiv2010

On the A-Obstacle Problem and the Hausdorff Measure of its Free Boundary

In this paper we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L^1 data. We also extend the Lewy-Stampacchia inequalities to the general framework of L^1 data, and show convergence and stability results. We then prove that the free boundary has finite N-1 Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p>2.

math.AP↗

arXiv2010

On the Two Obstacles Problem in Orlicz-Sobolev Spaces and Applications

We prove the Lewy-Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone operators. As a consequence for a general class of quasi-linear elliptic operators of Ladyzhenskaya-Uraltseva type, including p(x)-Laplacian type operators, we derive new results of $C^{1,α}$ regularity for the solution. We also apply those inequalities to obtain new results to the N-membranes problem and the regularity and monotonicity properties to obtain the existence of a solution to a quasi-variational problem in (generalized) Orlicz-Sobolev spaces.

math.AP↗

arXiv2007

On a constrained reaction-diffusion system related to multiphase problems

We solve and characterize the Lagrange multipliers of a reaction-diffusion system in the Gibbs simplex of R^{N+1} by considering strong solutions of a system of parabolic variational inequalities in R^N. Exploring properties of the two obstacles evolution problem, we obtain and approximate a N-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear reaction terms. We also show continuous dependence results and we establish sufficient conditions of non-degeneracy for the stability of those phase subregions.

math.AP↗