On the First Curve of the Fucik Spectrum of an Elliptic Operator with Weight

For the fully nonlinear uniformly elliptic equations,it is well-known that the solutions are of C~(2,α) if the nonlinear operators are convex(or concave).By weakening this hypothesis to that the nonlinear operators are of local C~(1, β) almost everywhere for 0 β1, the C~(2,α) regularity of the solutions is still shown in this paper.To prove the conclusion,a kind of iteration method is designed where the Holder continuity is measured by L~p norms.

Classical Solutions of Fully Nonlinear Elliptic Equations

Fil: Da Silva, Joao Vitor. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms.12161

Regularity properties for p−dead core problems and their asymptotic limit as p→∞

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type (
 $$1< p< \infty $$
 ) with strong absorption condition: $$\begin{aligned} -\mathrm {div}(\Phi (x, u, \nabla u)) + \lambda _0(x) u_{+}^q(x) = 0 \quad \hbox {in} \quad \Omega \subset \mathbb {R}^N, \end{aligned}$$
 where $$\Phi : \Omega \times \mathbb {R}_{+} \times \mathbb {R}^N \rightarrow \mathbb {R}^N$$
 is a vector field with an appropriate p-structure, $$\lambda _0$$
 is a non-negative and bounded function and $$0\le q<p-1$$
 . Such a model permits existence of solutions with dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. We establish sharp and improved $$C^{\gamma }$$
 regularity estimates along free boundary points, namely $$\mathfrak {F}_0(u, \Omega ) = \partial \{u>0\} \cap \Omega $$
 , where the regularity exponent is given explicitly by $$\gamma = \frac{p}{p-1-q} \gg 1$$
 . Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of $$(N-1)$$
 -Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the p-Laplace operator $$-\Delta _p u + \lambda _0 u^q\chi _{\{u>0\}} = 0$$
 for any $$\lambda _0>0$$
 .

Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

C1,α regularity for fully nonlinear elliptic equations with superlinear growth in the gradient

We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form Eq
 $$\begin{aligned} u_t - F\left( D^2u, Du, X, t\right) = f(X,t) \quad \text{ in } \quad Q_1, \end{aligned}$$
 where F is elliptic with respect to the Hessian argument and $$f \in L^{p,q}(Q_1)$$
 . The quantity $$\Xi (n, p, q) := \frac{n}{p}+\frac{2}{q}$$
 determines to which regularity regime a solution of (Eq) belongs. We prove that when $$1< \Xi (n,p,q) < 2-\epsilon _F$$
 , solutions are parabolically $$\alpha $$
 -Holder continuous for a sharp, quantitative exponent $$0< \alpha (n,p,q) < 1$$
 . Precisely at the critical borderline case, $$\Xi (n,p,q)= 1$$
 , we obtain sharp parabolic Log-Lipschitz regularity estimates. When $$0< \Xi (n,p,q) <1$$
 , solutions are locally of class $$C^{1+ \sigma , \frac{1+ \sigma }{2}}$$
 and in the limiting case $$\Xi (n,p,q) = 0$$
 , we show parabolic $$C^{1, \text {Log-Lip}}$$
 regularity estimates provided F has “better” a priori estimates.

Sharp regularity estimates for second order fully nonlinear parabolic equations

We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form \begin{equation}\label{Meq}\tag{Eq} u_t- F(D^2u, Du, X, t) = f(X,t) \quad \mbox{in} \quad Q_1, \end{equation} where $F$ is elliptic with respect to the Hessian argument and $f \in L^{p,q}(Q_1)$. The quantity $\kappa(n, p, q):=\frac{n}{p}+\frac{2}{q}$ determines to which regularity regime a solution of \eqref{Meq} belongs. We prove that when $1< \kappa(n,p,q) < 2-\epsilon_F$, solutions are parabolic-H\"{o}lder continuous for a sharp, quantitative exponent $0< \alpha(n,p,q) < 1$. Precisely at the critical borderline case, $\kappa(n,p,q)= 1$, we obtain sharp Log-Lipschitz regularity estimates. When $0< \kappa(n,p,q) <1$, solutions are locally of class $C^{1+ \sigma, \frac{1+ \sigma}{2}}$ and in the limiting case $\kappa(n,p,q) = 0$, we show $C^{1, \text{Log-Lip}}$ regularity estimates provided $F$ has "better" \textit{a priori} estimates.

João Vitor da Silva

Papers

Sharp regularity estimates for second order fully nonlinear parabolic equations

Regularity properties for p−dead core problems and their asymptotic limit as p→∞

Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

Sharp regularity estimates for second order fully nonlinear parabolic equations

Regularity for degenerate evolution equations with strong absorption