Gradient Of Weak Solutions Research Articles

Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem

We prove the local Hölder regularity of weak solutions to the mixed local nonlocal parabolic equation of the formut−Δu+P.V.∫Rnu(x,t)−u(y,t)|x−y|n+2sdy=0, where 0<s<1; for every exponent α0∈(0,1). Here, Δ is the usual Laplace operator. Next, we show that the gradients of weak solutions are also α-Hölder continuous for some α∈(0,1). Our approach is purely analytic and it is based on perturbation techniques.

Optimal Regularity for Elliptic Equations With Measurable Nonlinearities Under Nonstandard Growth

Abstract We are concerned with weak solutions of elliptic equations involving measurable nonlinearities with Orlicz growth to address what would be the weakest regularity condition on the associated nonlinearity for the Calderón–Zygmund theory. We prove that the gradient of weak solution is as integrable as the nonhomogeneous term under the assumption that the nonlinearity is only measurable in one of the variables while it has a small BMO assumption in the other variables. To this end, we develop a nonlinear Moser-type iteration argument for such a homogeneous reference problem with one variable–dependent nonlinearity under Orlicz growth to establish $W^{1,q}$–regularity for every $q&amp;gt;1$. Our results open a new path into the comprehensive understanding of the problem with nonstandard growth in the literature of optimal regularity theory in highly nonlinear elliptic and parabolic equations.

Regularity for a Nonlinear Discontinuous Subelliptic System with Drift on the Heisenberg Group

In this paper, we prove the partial Hölder regularity of weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions to a nonlinear discontinuous subelliptic system with drift on the Heisenberg group by theA-harmonic approximation, where the coefficients in the nonlinear subelliptic system are discontinuous and satisfy the VMO condition forx, ellipticity and growth condition with the growth index1&lt;p&lt;2for the Heisenberg gradient variable, and the nonhomogeneous terms satisfy the controllable growth condition and the natural growth condition, respectively.

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.

Weighted Lorentz estimates for nonlinear elliptic obstacle problems with partially regular nonlinearities

A global estimate in the framework of weighted Lorentz spaces is reached for the gradient of weak solution to a class of nonlinear elliptic obstacle problems with partially regular nonlinearities in a bounded Reifenberg flat domain. We mainly assume that the nonlinearities are merely measurable in one spatial variable and have small BMO seminorms in the remaining variables and that the underlying domain is flat in the sense of Reifenberg. As an application, we also present a global Lorentz estimate for the gradients of weak solutions to Dirichlet problems of relevant nonlinear elliptic equations under controlled growth in Reifenberg domains based on the bootstrap argument.

Interior regularity of obstacle problems for nonlinear subelliptic systems with VMO coefficients

This article is concerned with an obstacle problem for nonlinear subelliptic systems of second order with VMO coefficients. It is shown, based on a modification of A-harmonic approximation argument, that the gradient of weak solution to the corresponding obstacle problem belongs to the Morrey space L_{X,mathrm{loc}}^{2,lambda }.

Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions

This paper studies regularity estimates in Lebesgue spaces for gradients of weak solutions of a class of general quasilinear equations of p-Laplacian type in bounded domains with inhomogeneous conormal boundary conditions. In the considered class of equations the leading terms are vector-valued functions that are measurable in the x-variable and that depend in a nonlinear way on the solution and on its gradient. This class of equations consists of the well-known class of degenerate p-Laplace equations for $$p >1$$ . Under some sufficient conditions, we establish local interior, local boundary, and global $$W^{1,q}$$ -regularity estimates for weak solutions with $$q>p$$ , assuming that the weak solutions are in the John–Nirenberg BMO space. The paper therefore improves available results because it removes the boundedness or continuity assumptions on solutions. Our results also unify and cover known results for equations in which the principals are only allowed to depend on x-variable and gradient of solution variable. More than that, this paper gives a method to treat non-homogeneous boundary value problems directly without using any form of translations that is sometimes complicated due to the nonlinearities.

Local Hölder estimates for non-uniformly variable exponent elliptic equations in divergence form

In this paper we obtain the local Hölder regularity of the gradients of weak solutions for a class of non-uniformly nonlinear variable exponent elliptic equations in divergence formincluding the following special modelunder some proper assumptions on Ai and the Hölder continuous functions f, pi(x) for i = 1, 2.

Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients

We prove global Lorentz estimates for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coefficients over a bounded nonsmooth domain. Here, we assume that the leading coefficients are measurable in one variable and have small BMO semi-norms in the other variables, variable exponents p(x) satisfy log-Hölder continuity, and the boundaries of domains are so-called Reifenberg flat. This is a natural outgrowth of the classical Calderón-Zygmund estimates to a variable power of the gradient of weak solutions in the scale of Lorentz spaces for such variational inequalities beyond the Lipschitz domain.

On Morrey and BMO regularity for gradients of weak solutions to nonlinear elliptic systems with non-differentiable coefficients

On Morrey and BMO regularity for gradients of weak solutions to nonlinear elliptic systems with non-differentiable coefficients

Smoothness of the gradient of weak solutions of degenerate linear equations

Let Q(x) be a nonnegative definite, symmetric matrix such that \(\sqrt {Q\left( x \right)} \) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q∇u) = b, we find sufficient conditions in order that \(\sqrt Q \nabla u\) has some first order smoothness. Specifically, if Ω is a bounded open set in Rn, we study when the components of \(\sqrt Q \nabla u\) belong to the first order Sobolev space WQ1,2(Ω) defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of Σi=1nXi′Xiu + b = 0. We do not assume that {Xi} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.

Optimal partial regularity for nonlinear sub-elliptic systems with Dini continuous coefficients in Carnot groups

This paper is concerned with the interior regularity for nonlinear sub-elliptic systems with Dini continuous coefficients under superquadratic controllable growth conditions in Carnot groups. We adapt the technique of the $\mathcal{A}$ -harmonic approximation to the case of sub-elliptic systems in divergence form, and we show a partial regularity result for weak solutions. In particular, our result is optimal in the sense that in the case of Holder continuous coefficients we obtain directly the optimal Holder exponent for the horizontal gradient of weak solutions on its regular set.

Hölder continuity for nonlinear sub-elliptic systems with sub-quadratic growth

This paper is concerned with partial regularity of weak solutions to nonlinear sub-elliptic systems under sub-quadratic natural growth conditions. We begin with establishing a Sobolev-Poincare type inequality associated with Hormander’s vector fields for $$u\in HW^{1,m}(\Omega , \mathbb {R}^N)$$ with $$1<m<2$$ . Then $$\mathcal {A}$$ -harmonic approximation method is applied, and partial Holder continuity with optimal local Holder exponent for gradients of weak solutions to the systems is established.

Optimal partial regularity for sub-elliptic systems related to Hörmander’s vector fields: the controllable growth case

This paper is devoted to partial regularity of weak solutions to nonlinear sub-elliptic systems associated with Hormander’s vector fields with \(p\) growth for \(p\ge 2\). Using \(\mathcal A \)-harmonic approximation technique introduced by Duzaar and Steffen, we prove Holder continuity for gradient of weak solutions with optimal local Holder exponent.

Regularity theory of the gradient for general nonlinear parabolic systems

In this article, we prove partial Hölder regularity of the spatial gradient of weak solutions to a nonlinear parabolic system of second order in divergence form in the case of quadratic growth in the gradient.

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally Holder continuous with Holder exponent $${\beta \in (0,1)}$$ with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is Holder continuous with exponent β with respect to the parabolic metric.

A note on the global regularity of steady flows of generalized Newtonian fluids

We establish regularity results for solutions of a generalized Newtonian model in a cubic domain. We prove regularity results in the L2-space for the second derivatives of the velocity and the first derivatives of the pressure. Further, we show that the gradient of weak solutions is integrable up to the boundary with any finite exponent.

Bifurcation phenomena associated to a class of p -Laplacian like operators

Global bifurcation of positive solutions for some degenerate quasilinear elliptic problems is considered. The uniform estimate of the gradient of weak solutions is given. This estimate is crucial in our arguments.

Everywhere Hölder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ∂u i∂t−D α A i α (∇u)=B i(∇u) in (i=1,...,) (Q=Ω×(0,T), R n a domain), where the functionsB i may have a quadratic growth. Under the assumptionsn≤2 and ∇u ɛL loc 4+δ (Q; R nN ) (δ>0) we prove that ∇u is locally Holder continuous inQ.