Gradient Estimates For Solutions Research Articles

Existence and regularity estimates for quasilinear equations with measure data : the case 1 < p ≤ (3n − 2)∕(2n − 1)

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-{\rm div} (|\nabla u|^{p-2} \nabla u)= \delta\, |\nabla u|^q +\mu$ in a bounded main $\Om\subset\RR^n$ potentially with non-smooth boundary. Here either $\delta=0$ or $\delta=1$, $\mu$ is a finite signed Radon measure in $\Omega$, 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 $\delta=1$ which corresponds to a Riccati type equation, we settle the question of solvability that has been raised for some time in the literature.

Global gradient estimates for very singular quasilinear elliptic equations with non-divergence data

This paper continues the development of regularity results for quasilinear elliptic equations −div(A(x,∇u))=μinΩ,andu=0on∂Ω,in Lorentz and Lorentz–Morrey spaces, where Ω⊂Rn (n≥2); A is a monotone Carathéodory vector valued operator acting between W01,p(Ω) and its dual W−1,p′(Ω); and μ is a datum in some Lebesgue space Lm(Ω), for m<p′. It emphasizes that in this paper, we restrict our study to the case of ‘very singular’ when 1<p≤3n−22n−1, and under mild assumption that the p-capacity uniform thickness condition is imposed on the complement of domain Ω. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz–Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the ‘very singular’ case still remains a challenge, specifically related to Lorentz and Lorentz–Morrey spaces.

A regularity result via fractional maximal operators for p-Laplace equations in weighted Lorentz spaces

ABSTRACT We study the global regularity in weighted Lorentz spaces for the gradient of weak solutions to p-Laplace equations in an open-bounded non-smooth domain which satisfies a p-capacity condition. Moreover, the gradient estimate is presented in terms of fractional maximal operators which are related to the Riesz potential and the fractional derivative. Our main concern is to extend the results of Tran and Nguyen [New gradient estimates for solutions to quasi-linear divergence form elliptic equations with general Dirichlet boundary data. J Differ Equ. 2020;268(4):1427–1462] but in the weighted Lorentz spaces associated to the Muckenhoupt weights. The key technique that we use in this paper is the good-λ inequality and the interesting properties of the cut-off fractional maximal operators.

New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions:{div(A(x,∇u))=div(|F|p−2F)inΩ,u=σon∂Ω, where Ω⊂Rn (n≥2), the nonlinearity A is a monotone Carathéodory vector valued function defined on W01,p(Ω) for p>1 and the p-capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F∈Lp(Ω;Rn), the problem is set up with general Dirichlet boundary data σ∈W1,p(Ω). In this paper, the optimal good-λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45,52,53] and references therein.

Localized elliptic gradient estimate for solutions of the heat equation on $${ RCD}^*(K,N)$$RCD∗(K,N) metric measure spaces

In this paper, we will establish a localized elliptic gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels.

Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: in a bounded domain , under minimal regularity assumptions on the boundary of domain and on nonlinearity A. Then results yields existence of a solution to the Riccati type parabolic equations: where and is a bounded Radon measure.

Cauchy problem for doubly singular parabolic equation with gradient source

This paper examines the Cauchy problem for doubly singular parabolic equation with a source term depending solely on the gradient. We establish the local and global existence of solutions when initial data is merely a function in (). Moreover, the uniform -estimates and gradient estimates of solutions are obtained.

A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature

We consider a singular or degenerate elliptic problem in a proper domain and we prove a gradient bound and some symmetry results.

Gradient estimate for solutions to Poisson equations in metric measure spaces

Let ( X , d ) be a complete, pathwise connected metric measure space with a locally Ahlfors Q-regular measure μ, where Q > 1 . Suppose that ( X , d , μ ) supports a (local) ( 1 , 2 ) -Poincaré inequality and a suitable curvature lower bound. For the Poisson equation Δ u = f on ( X , d , μ ) , Moser–Trudinger and Sobolev inequalities are established for the gradient of u. The local Hölder continuity with optimal exponent of solutions is obtained.

Global solutions of inhomogeneous Hamilton-Jacobi equations

We consider the viscous Hamilton-Jacobi (VHJ) equationu t-Δu=|∇u|p+h(x). For the Dirichlet problem withp>2, it is known thatgradient blow-up may occur in finite time (on the boundary). Whereas considerable effort has been devoted to study the large time behavior of solutions of the equationu t-Δu=g(x,u), whereamplitude blow-up may occur if for instanceg(x,u)≈u p asu→∞ andp>1, relatively little is known in the case of (VHJ). The aim of this paper is to investigate this question. More precisely, we study the relations between and we obtain a precise description of the global dynamics for (VHJ). Namely, we show that (i) implies (ii) and that in this case, all global solutions converge uniformly to the (unique) stationary solution. In the radial case, we prove that, conversely, (ii) implies (i). Moreover, for certain (smooth) functionsh, we obtain the existence of global classical solutions with gradient blowing up in infinite time. For 1p-2 or for the Cauchy problem, all solutions are global, but we establish similar relations between the existence of bounded or locally bounded solutions and the existence of stationary solutions. Our proofs depend on some new gradient estimates of solutions, local and global in space, obtained by Bernstein type arguments. As another consequence of these estimates we prove a parabolic Liouville-type theorem for solutions ofu t\t-Δu=│Δu│p in ℝNx(\t-\t8,0). Various other results are obtained, including universal bounds for global solutions.

Quasilinear Two-Phase Venttsel Problems

A majorant of the Holder constant is established for solutions of the two-phase quasilinear parabolic or elliptic boundary-value problems with degenerate and nondegenerate Venttsel conditions on an interface. In addition, gradient estimates for solutions of the two-phase quasilinear nondegenerate Venttsel problems are obtained, and the solvability in Sobolev and Holder spaces is proved. Bibliography: 14 titles.

Lower bound gradient estimates for solutions of schrodinger equations and heat kernels

Lower bound gradient estimates for solutions of schrodinger equations and heat kernels

On the quasilinear stationary Ventzel boundary-value problem

A priori estimates for gradients of solutions of a boundary-value problem for a quasilinear nondivergent elliptic equation with the quasilinear Ventzel boundary condition are established. By these estimates, existence theorems in the Hölder and Sobolev spaces are proved. Bibliography:11 titles.

A remark on a system of inequalities with bilateral obstacles

up+‘(x) k s up(x) I up+l(x) + K, x E Sz APUP = fP if up+‘(x) k < up(x) < up+‘(x) + K, x E Q APuP 5 fP if up(x) = up+l(x) + K, x E l2 (1.1) APuP L f if up(x) = up+‘(x) k, x E Q u”(x) = 0 onaQ,p= 1,2 ,..., m. where p ranges over (1, . . . , m) and urn+’ represents u’, is dealt with by the second author in [5], and the existence of a weak solution of (1.1) is established under the assumption: AP’s have sufficiently large zeroth order coefficients. (1.2) The proof of the existence result in [5] is based on the IV’*” estimate, independent of E, for the solution u, = 124:. . . . , u,“) of the following approximate system: APu, + &(u,p uf+l K) &(u;+~ k uf) = f in R, (1.3) u,” = 0 0naQ,p = 1,2 ,..., m, where p ranges over ( 1, . . . , ml, P, is the so-called penalty function and uF+’ represents u:. The gradient estimate for solutions U: of (1.3) in [5] is obtained by using Bernstein’s method, and assumption (1.2) is solely employed to make it possible. In addition, it should be remarked that standard variants of the Bernstein trick like in [2] and [4] which are useful to obtain the IV**” estimate for solutions of the system of variational inequalities with unilateral constraints without (1.2) do not work in our case.

Local estimates for gradients of solutions of non‐uniformly elliptic and parabolic equations

Communications on Pure and Applied MathematicsVolume 23, Issue 4 p. 677-703 Article Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations O. A. Ladyzhenskaya, O. A. Ladyzhenskaya Leningrad UniversitySearch for more papers by this authorN. N. Ural'Tseva, N. N. Ural'Tseva Leningrad UniversitySearch for more papers by this author O. A. Ladyzhenskaya, O. A. Ladyzhenskaya Leningrad UniversitySearch for more papers by this authorN. N. Ural'Tseva, N. N. Ural'Tseva Leningrad UniversitySearch for more papers by this author First published: July 1970 https://doi.org/10.1002/cpa.3160230409Citations: 99AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume23, Issue4July 1970Pages 677-703 RelatedInformation