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Diego Moreira

Publications and source records attributed to Diego Moreira.

6 recordsLinked to original sources
arXiv2025

A quantitative Hopf-Oleinik lemma for degenerate fully nonlinear operators and applications to free boundary problems

We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of $$|\nabla u|^αF(D^{2}u)=f $$ and, more generally, for viscosity supersolutions of $|\nabla u|^α\,{M}^-_{λ,Λ}(D^{2}u)\le f$. The result yields linear boundary growth with universal constants depending only on the structural data. We also exhibit a counterexample showing that the Hopf lemma fails for equations that act only in the large-gradient regime (in the sense of Imbert and Silvestre), thereby delineating the scope of our theorem. As applications, we obtain Lipschitz regularity for viscosity solutions of one-phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive $\varepsilon$-uniform Lipschitz bounds for a one-phase flame propagation model.

math.AP↗
arXiv2022

Tangential contact of free boundaries and the fixed boundary for variational solutions to a free transmission Problem

In this article we study functionals of the following type $$ \int_Ω \Big ( \langle A(x,u)\nabla u, \nabla u\rangle + Λ(x,u) \Big )\,dx $$ here $A(x,u)= A_+(x)χ_{\{u>0\}}+A_-(x) χ_{\{u\leq 0\}}$ for some elliptic and bounded matrices $A_{\pm}$ with Hölder continuous entries and $Λ(x,u) = λ_+(x) χ_{\{u>0\}} + λ_-(x) χ_{\{u\le 0\}}$. We prove that the free boundaries of minimizers of the above functional touches the fixed boundary $\partial Ω$ in a tangential fashion, provide the graph of boundary data touches its zeros smoothly. This assumption is reflected in the \eqref{DPT} condition.

math.AP↗
arXiv2022

Optimal Regularity in Transmission Problems]{Optimal regularity for variational solutions of free transmission problems

In this article we study functionals of the type considered in \cite{HS21}, i.e. $$ J(v):=\int_{B_1} A(x,u)|\nabla u|^2 +f(x,u)u+ Q(x)λ(u)\,dx $$ here $A(x,u)= A_+(x)χ_{\{u>0\}}+A_-(x) χ_{\{u<0\}}$, $f(x,u)= f_+(x)χ_{\{u>0\}}+f_-(x) χ_{\{u<0\}}$ and $λ(x,u) = λ_+(x) χ_{\{u>0\}} + λ_-(x) χ_{\{u\le 0\}}$. We prove the optimal $C^{0,1^-}$ regularity of minimizers of the functional indicated above (with precise Hölder estimates) when the coefficients $A_{\pm}$ are continuous functions and $μ\le A_{\pm}\le \frac{1}μ$ for some $0<μ<1$, with $f \in L^N(B_1)$ and $Q$ bounded. We do this by presenting a new compactness argument and approximation theory similar to the one developed by L. Caffarelli in \cite{Ca89} to treat the regularity theory for solutions to fully nonlinear PDEs. Moreover, we introduce the $\mathcal{T}_{a,b}$ operator that allows one to transfer minimizers from the transmission problems to the Alt-Caffarelli-Friedman type functionals, {in small scales,} allowing this way the study of the regularity theory of minimizers of Bernoulli type free transmission problems.

math.AP↗
arXiv2016

Inhomogeneous Hopf-Oleĭnik Lemma and Applications. Part IV: Sharp Krylov Boundary Gradient Type Estimates for Solutions to Fully Nonlinear Differential Inequalities with unbounded coefficients and $C^{1,Dini}$ boundary data

In this paper we provide another application of the Inhomogeneous Hopf-Ole\uınik Lemma (IHOL) proved in \cite{BM-IHOL-PartI} or \cite{Boyan-2}. As a matter of fact, we also provide a new and simpler proof of a slightly weaker version IHOL for the uniformly elliptic fully nonlinear case which is sufficient for most purposes. The paper has essentially two parts. In the first part, we use IHOL for unbounded RHS to develop a Caffarelli's "Lipschitz implies $C^{1,α}$" approach to prove Ladyzhenskaya-Uraltseva boundary gradient type estimates for functions in $S^{*}(γ, f)$ that vanishes on the boundary. Here, unbounded RHS means that $f\in L^{q}$ with $q>n$. This extends the celebrated Krylov's boundary gradient estimate proved in \cite{Krylov}. A Phragmén-Lindelöf classification result for solutions in half spaces is recovered from these estimates. Moreover, a Hölder estimate up to the boundary (in the half-ball) for $u(x)/x_{n}$ is obtained. In the second part, we extend the previous results for functions in $S^{*}(γ, σ, f)$ where $γ,f\in L^{q}$ with $q>n$ that have a $C^{1,Dini}$ boundary data on a $W^{2,q}$ domain. Here, we use an "improvement of flatness" strategy suited to the unbounded coefficients scenario. As a consequence of that, a quantitative version of IHOL under pointwise $C^{1,Dini}$ boundary regularity is obtained.

math.AP↗
arXiv2008

On the regularity of maximal operators

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.

math.CA↗