We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is smooth except at a finite set of conical singularities. This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen. Then we get existence results among all metrics on surfaces of a given genus, leading to the existence of minimal isometric immersions of smooth compact Riemannian manifold $(M,g)$ of dimension 2 into some $k$-sphere by first eigenfunctions. At last, we also answer a conjecture of Friedlander and Nadirashvili which asserts that the supremum of the first eigenvalue of the Laplacian on a conformal class can be taken as close as we want of its value on the sphere on any orientable surface.
In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical $p$-Laplacian situation, namely $C^{s+α}_{loc}$-regularity for the homogeneous equation.
The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.
We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces.
We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
We prove the local gradient Hölder regularity of viscosity solutions to the inhomogeneous normalized $p(x)$-Laplace equation $$ -Δu-(p(x)-2)\frac{\left\langle D^{2}uDu,Du\right\rangle }{\left|Du\right|^{2}} = f(x), $$ where $p$ is Lipschitz continuous, $\inf p>1$, and $f$ is continuous and bounded.
In this note, we prove the boundary Hölder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The key idea, following [3], is that the strong maximum principle and the scaling invariance imply the boundary Hölder regularity.
In this paper, we study the second order Sobolev regularity of solutions to the parabolic $p$-Laplace equation. For any $p$-parabolic function $u$, we show that $D(|Du|^{\frac{p-2+s}{2}}Du)$ exists as a function and belongs to $L^{2}_{\text{loc}}$ with $s>-1$ and $1<p<\infty$. The range of $s$ is sharp.
We consider interior Hölder regularity of the spatial gradient of viscosity solutions to the normalized $p(x,t)$-Laplace equation $$ u_t=\left(δ_{ij}+(p(x,t)-2)\frac{u_i u_j}{|Du|^2}\right)u_{ij} $$ with some suitable assumptions on $p(x,t)$, which arises naturally from a two-player zero-sum stochastic differential game with probabilities depending on space and time.
In this paper, we obtain local Hölder regularity for bounded, weak solutions to the anisotropic $p$-Laplace equation whose prototype structure is given by $$ \sum_{i=1}^N (|u_{x_i}|^{p_i-2}u_{x_i})_{x_i}=0,$$ where $1 < p_1 \leq p_2 \leq \cdots \leq p_N < \infty$. Under an additional assumption that double truncates of the solution are sub/super solution, we obtain Hölder regularity for the anisotropic equation.
We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-Δ_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $Δ_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the nonlocal $p$-Laplace operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi-Nash-Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.
We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. This work can be viewed as the nonlocal counterpart to the ideas developed by Tiziano Granucci.
We study existence and regularity properties of solutions to the singular $p$-Laplacean parabolic system in a bounded domain $Ω$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(Ω))$, $\varepsilon\geq0$, integrability properties of the second spatial derivatives and of the time derivative of the solutions. Hence, for suitable $p$ and exponents $r,\,q$, by Sobolev embedding theorems, we deduce global regularity of $u$ and $\nabla u$ in Hölder spaces. Finally we prove a global pointwise bound for the solution under the assumption $p>\frac{2n}{n+2}$.
Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.
We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,Ω):=\int_Ω\Big( |\nabla u(x)|^p+χ_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous.
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.
We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation $s$ reaches $1$.
Second-order estimates are established for solutions to the $p$-Laplace system with right-hand side in $L^2$. The nonlinear expression of the gradient under the divergence operator is shown to belong to $W^{1,2}$, and hence to enjoy the best possible degree of regularity. Moreover, its norm in $W^{1,2}$ is proved to be equivalent to the norm of the right-hand side in $L^2$. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local esti…
We consider quasilinear parabolic equations with measurable coefficients when the right-hand side is a signed Radon measure with finite total mass, having $p$-Laplace type: $$u_t - \textrm{div} \, \mathbf{a}(Du,x,t) = μ\quad \textrm{in} \ Ω\times (0,T) \subset \mathbb{R}^n \times \mathbb{R}.$$ In the singular range $\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$, we establish regularity estimates for the spatial gradient of solutions in the Marcinkiewicz spaces, under a suitable density condition of the right-hand side measure.
We show that local weak solutions to parabolic systems of p-Laplace type are H{ö}lder continuous in time with values in a spatial Lebesgue space and H{ö}lder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.
We study the regularity of the extremal solution $u^*$ to the singular reaction-diffusion problem $-Δ_p u = λf(u)$ in $Ω$, $u =0$ on $\partial Ω$, where $1<p<2$, $0 < λ< λ^*$, $Ω\subset \mathbb{R}^n$ is a smooth bounded domain and $f$ is any positive, superlinear, increasing and (asymptotically) convex $C^1$ nonlinearity. We provide a simple proof of known $L^r$ and $W^{1,r}$ \textit{a priori} estimates for $u^*$, i.e. $u^* \in L^\infty(Ω)$ if $n \leq p+2$, $u^* \in L^{\frac{2n}{n-p-2}}(Ω)$ if $n > p+2$ and $|\nabla u^*|^{p-1} \in L^{\frac{n}{n-(p'+1)}} (Ω)$ if $n > p p'$.
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Hölder regularity under proper geometric conditions. "Unified" means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the $p-$Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the meas…
We obtain optimal regularity in the Sobolev space $W_0^{1,τ}(Ω)$ for the unique solution of $$ -Δ_m u=K(x)u^{-p} \mbox{in} Ω, \quad u=0\mbox{on}\partial Ω. $$ Here $Ω\subset{\mathbb R}^N$ is a smooth and bounded domain, $m>1$, $p\geq 0$ and $K\in C(Ω)$ is a positive function that behaves like ${\rm dist}(x,\partialΩ)^{-q}$ for some $q\geq 0$ with $p+q< 2-\frac{1-p}{m}$. We obtain that the unique weak solution to the above problem belongs to $W_0^{1,τ}(Ω)$ for $$ m\leq τ<\frac{m+p-1}{p+q-1}\quad \mbox{if}p+q>1, $$ and $$ m\leq τ<\infty\quad \mbox{if}p+q=1. $$ The above range of $τ$ is optimal.
Perron's method and Wiener's criterion have entirely solved the Dirichlet problem for the Laplace equation. Since then, this approach has attracted the attention of many mathematicians for applying these ideas in the more general equations. So, in this paper, we extend the Perron method and the Wiener criterion to the $G(\cdot)$-Laplace equation.
Hölder regularity for the parabolic perturbed fractional 1-Laplace equations
Harbin Institute of Technology（哈尔滨工业大学）；School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
In a paper from 1996, D. Jerison and C. Kenig among other results provided a $H^{1/2}$ regularity result for the Dirichlet problem for the Laplace equation in Lipschitz domains. In this article, we adopt a Hilbertian approach to construct two Riesz bases for $L^2(\partial Ω),$ which will allow to find in a different way some of the results of D. Jerison and C. Kenig, and G. Savaré (1998) about the regularity issue of the Laplace equation.
We announce some new results for proving Hölder continuity of weak solutions to quasilinear parabolic equations whose prototype takes the form $$u_t - div (|\nabla u|^{p-2}\nabla u)= 0 \qquad \text{or} \qquad u_t - div (|u_{x_1}|^{p_1-2}u_{x_1},|u_{x_2}|^{p_2-2}u_{x_2},\ldots |u_{x_N}|^{p_N-2}u_{x_N})=0$$ and $1<\{p_1,p_2,\ldots,p_N\}<\infty$. We develop a new technique which is independent of the "method of intrinsic scaling" developed by E.DiBenedetto in the degenerate case ($p\geq 2$) and E.DiBenedetto and Y.Z.Chen in the singular case ($p\leq 2$) and instead uses a new and elementary linea…
In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \textit{a priori} estimates for semi-stable solutions of $-Δ_p u= g(u)$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^\infty(Ω)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(Ω)\…
This paper concerns value functions of time-dependent tug-of-war games. We first prove the existence and uniqueness of value functions and verify that these game values satisfy a dynamic programming principle. Using the arguments in the proof of existence of game values, we can also deduce asymptotic behavior of game values when $T \to \infty$. Furthermore, we investigate boundary regularity for game values. Thereafter, based on the regularity results for value functions, we deduce that game values converge to viscosity solutions of the normalized parabolic $p$-Laplace equation.
In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$ time differentiability in an expontential Besov-Orlicz space for the solution process $u$. Furthermore, we prove $1/2$ time differentiability of the nonlinear gradient $|\nabla u|^{\frac{p-2}{2}} \nabla u$ in a Nikolskii space.
We consider systems of stochastic evolutionary equations of the $p$-Laplace type. We establish convergence rates for a finite-element based space-time approximation, where the error is measured in a suitable quasi-norm. Under natural regularity assumptions on the solution, our main result provides linear convergence in space and convergence of order $α$ in time for all $α\in(0,\frac{1}{2})$. The key ingredient of our analysis is a random time-grid, which allows us to compensate for the lack of time regularity. Our theoretical results are confirmed by numerical experiments.
We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau type. By the use of special weak different quotients, adapted to the variational structure we bound fractional derivatives of $u_t$ in time and space direction. Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic p-Laplace equation.
The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the Dirichlet energy among homeomorphisms often leads to mappings which are neither harmonic nor homeomorphisms. We prove that such mappings are harmonic outside of a singular set with small image. On the singular set they are locally Lipschitz, but not necessarily differentiable.
