Search results for " 35D40"
showing 6 items of 6 documents
Local regularity for quasi-linear parabolic equations in non-divergence form
2018
Abstract We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the p -Laplacian type and in non-divergence form. We provide local Holder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Holder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.
The tusk condition and Petrovski criterion for the normalized $p\mspace{1mu}$-parabolic equation
2017
We study boundary regularity for the normalized $p\mspace{1mu}$-parabolic equation in arbitrary bounded domains. Effros and Kazdan (Indiana Univ. Math. J. 20 (1970), 683-693) showed that the so-called tusk condition guarantees regularity for the heat equation. We generalize this result to the normalized $p\mspace{1mu}$-parabolic equation, and also obtain H\"older continuity. The tusk condition is a parabolic version of the exterior cone condition. We also obtain a sharp Petrovski criterion for the regularity of the latest moment of a domain. This criterion implies that the regularity of a boundary point is affected if one side of the equation is multiplied by a constant.
Equivalence of viscosity and weak solutions for the $p(x)$-Laplacian
2010
We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem.
Equivalence of viscosity and weak solutions for the normalized $p(x)$-Laplacian
2018
We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$ regularity for the viscosity solutions of the normalized $p(x)$-Laplace equation. As an additional application, we prove a Rad\'o-type removability theorem.
Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian
2021
AbstractWe prove a local Hölder estimate for any exponent $0<\delta <\frac {1}{2}$ 0 < δ < 1 2 for solutions of the dynamic programming principle $$ \begin{array}{@{}rcl@{}} u^{\varepsilon} (x) = \sum\limits_{j=1}^{n} \alpha_{j} \underset{\dim(S)=j}{\inf} \underset{|v|=1}{\underset{v\in S}{\sup}} \frac{u^{\varepsilon} (x + \varepsilon v) + u^{\varepsilon} (x - \varepsilon v)}{2} \end{array} $$ u ε ( x ) = ∑ j = 1 n α j inf dim ( S ) = j sup v ∈ S | v | = 1 u ε ( x + ε v ) + u ε ( x − ε v ) 2 with α1,αn > 0 and α2,⋯ ,αn− 1 ≥ 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $…
Equivalence of viscosity and weak solutions for a $p$-parabolic equation
2019
AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .