We introduce the concept of kinetic maximal $L^p$-regularity with temporal weights and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterized in terms of anisotropic Besov spaces. We further extend the property of kinetic maximal $L^p_μ$-regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal $L^p_μ$-regularity can be used to obtain local existence of solutions to a class of quasilinear kinetic equations and illustrate our result with a quasilinear kinetic diffusion equation.
For a linear nonvariational operator structured on smooth Hörmander's vector fields, with Hölder continuous coefficients, we prove a regularity result in the spaces of Hölder functions. We deduce an analogous regularity result for nonvariational degenerate quasilinear equations.
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in R^n. Well-posedness of solutions reconstructing maximal parabolic regularity in nonlinear thermoelastic plates is established. In addition, exponential decay rates for strong solutions are also shown.
We prove the global regularity of weak solutions to the conormal derivative boundary value problem for quasilinear elliptic equations in divergence form on Lipschitz domains under the controlled growth conditions on the low order terms. The leading coefficients are in the class of BMO functions with small mean oscillations.
We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.
We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,α}$ local estimates for solutions of a degenerate equation in non divergence form.
In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in $H^{3}\times H^{2}$. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.
We establish sharp geometric $C^{1+α}$ regularity estimates for bounded weak solutions of evolution equations of $p$-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.
In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and $L^1$ data. In the same setting we study sharp (local and global) integral estimates for the second derivatives of the solutions.
We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.
This paper is concerned with a class of quasilinear elliptic equations involving some potentials related to the Caffarelli-Korn-Nirenberg inequality. We prove the local boundedness and Hölder continuity of weak solutions by using the classical De Giorgi techniques. Our result extends the results of Serrin \cite{Serrin64} and Corolado and Peral \cite{CP04}.
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for w…
The goal of this article is to establish local Lipschitz continuity of weak solutions for a class of degenerated elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity condition is a natural generalization of the p-Laplace equation and more general quasilinear elliptic equations with polynomial or exponential type growth.
We study a class of linearly coupled system of quasilinear equations. Under some assumptions on the nonlinear terms, we establish some results about the existence and regularity of vector solutions for the p-Laplacian systems by using variational methods. In particular, we get two pairs of nontrivial solutions. We also study their different asymptotic behavior of solutions as the coupling parameter tends to zero.
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in \cite{Zhong} of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\Hn$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of quasilinear pde's in $\Hn$ modelled on the equation $$\p_t u= \sum_{i=1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$
We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equati…
In this paper, we are interested in obtaining a unified approach for $C^{1,α}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman.
This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $Φ$-Laplacian operator. The proof of existence is based on a variant of the generalized Galerkin method that we developed inspired on ideas by Browder and a comparison principle. By using a kind of Moser iteration scheme we show $L^{\infty}(Ω)$-regularity for positive solutions
We consider inhomogeneous $p$-Laplace type equations of the form $-\mathrm{div}\left(a(\nabla u)\right)=f$ in a possibly anisotropic setting. Under general assumptions on the source term $f$, we obtain quantitative Sobolev regularity results for the stress field $a(\nabla u)$ and weighted $L^2$ estimates for the Hessian of the solution. As far as we know, our results are new or refine the ones available in literature also when restricted to the Euclidean setting.
We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous. More precisely, we show that it is Hölder continuous with respect to a certain metric associated to the operator. One of the essential features of this metric is that the metric balls are non doubling with respect to Lebesgue measure. The proof of the continuity together with a recent result by Rios et al. completes the result on hypoellipticity of a class o…
We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along $S(u)$, $u$ is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $S(u)$, $Du$ enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. Our main, leading result fosters a new understanding on smoothness properties of solutions to degenerate or sin…
In this paper, we obtain $C^{1,α}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switch…
This paper deals with the existence of positive solution for the singular quasilinear Schrödinger equation $-Δu -Δ(u^{2})u=h(x) u^{-γ} + f(x,u)~\mbox{in} ~ Ω,$ where $γ> 1$, $Ω\subset \mathbb{R}^{N}, (N\geq 3)$ is a bounded smooth domain, $0<h\in L^{1}(Ω)$, $f$ is a measurable function that can change signal and can be sublinear or has critical growth. Inspired by Sun \cite{Y} we derive a compatible condition on the couple $(h(x),γ)$, which is optimal for the existence of $H_{0}^{1}$-solution for this problem.
We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The…
It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $Φ$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-Δ_Φu= g(x,u), & \mbox{in}~Ω, u=0, & \mbox{on}~\partial Ω, \end{array} \right. \end{equation*} where $Δ_Φu :=\mbox{div}(ϕ(|\nabla u|)\nabla u)$ and $Ω\subset\mathbb{R}^{N}, N \geq 2,$ is a bounded domain with smooth boundary $\partialΩ$. Our work concerns on nonlinearities $g$ which can be homogeneous or non-homogeneous. For the homogeneous case we consider an existence result together with a re…
We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-{\rm div} (|\nabla u|^{p-2} \nabla u)= δ\, |\nabla u|^q +μ$ in a bounded main $\Om\subset\RR^n$ potentially with non-smooth boundary. Here either $δ=0$ or $δ=1$, $μ$ is a finite signed Radon measure in $Ω$, and $q$ is of linear or super-linear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. In particular, in the case $δ=1$ which corresponds to a Ricc…
It is established $L^{p}$ estimates for the fractional $Φ$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the Moser's iteration, we prove that any weak solution for fractional $Φ$-Laplacian operator defined in bounded domains belongs to $L^\infty(Ω)$ under appropriate hypotheses on the $N$-function $Φ$. Using the Orlicz space and taking into account the fractional setting for our problem the main results are stated for a huge class of nonlinear operators and nonlinearities…
We study regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in bounded domains in $\R^n$. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under a small BMO condition in $x$ for $\A$. As a consequence, we obtain that $\nabla u$ is in the classical Morrey space $\calM^{q,λ}$ or weighted space $L^q_w$ whenever $|\F|^{\frac{1}{p-1}}$ is respectively in $\calM^{q,λ}$ o…
In this paper study the regularity of continuous casting problem \begin{equation} \hbox{div}(|\nabla u|^{p-2}\nabla u-{\bf v} β(u))=0\tag{$\sharp$} \end{equation} for prescribed constant velocity $\bf v$ and enthalpy $β(u)$ with jump discontinuity at $u=0$. We establish the following estimates: local log-Lipschitz $p>2$ for $u$ (and BMO for $\nabla u$) for two phase, Lipschitz $p>1$ for one phase and linear growth up-to boundary near the contact points. We also prove that the free boundary is continuous curve in the direction of $\bf v$ in two spatial dimensions. The proof is based on a delica…
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…
This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -Δ_{p}u -Δ_{q}u = f(x) u^{-δ},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Cons…
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^1$-contraction property for the solutions.
In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder's fixed point theorem.
We consider an iterative procedure to solve quasilinear elliptic systems with $p$-growth. The scheme was first considered by Koshelev in the quadratic case $p=2$. We present numerical applications as well as applications to higher regularity properties.
In this article we prove local well-posedness in low-regularity Sobolev spaces for general quasilinear Schrödinger equations. These results represent improvements of the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove existence of solutions in very high regularity spaces. Our arguments here are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the authors.
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 diff…
We establish existence and regularity of positive solutions for a class of quasilinear elliptic systems with singular and superlinear terms. The approach is based on sub-supersolution methods for systems of quasilinear singular equations and the Schauder's fixed point Theorem.
In this paper we establish existence, nonexitence and regularity of positive solutions for a class of singular quasilinear elliptic systems subject to (super-) homogeneous condition. The approach is based on sub-supersolution methods for systems of quasilinear singular equations combined with perturbation arguments involving singular terms.
We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently small and regular, then there exists a unique global solution with optimal $L_p$-regularity. We show furthermore that the solution converges to zero at an exponential rate as time tends to infinity. Our techniques are based on maximal $L_p$-regularity for abstract quasilinear parabolic equations.
