Holder Continuity Of Weak Solutions Research Articles

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

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, local Holder continuity of weak solutions, 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.

$$ {\mathfrak{B}}_1 $$ classes of De Giorgi, Ladyzhenskaya, and Ural’tseva and their application to elliptic and parabolic equations with nonstandard growth

The article provides an application of the generalized De Giorgi functional classes to the proof of the HOlder continuity of weak solutions to quasilinear elliptic and parabolic equations with nonstandard growth conditions.

Regularity of Solutions to a Model Oblique Derivative Problem for Quasilinear Parabolic Systems with Nondiagonal Principal Matrices

We consider quasilinear parabolic systems of equations with nondiagonal principal matrices. The oblique derivative of a solution is defined on the plane part of the lateral surface of a parabolic cylinder. We do not assume smoothness of the principal matrix and the boundary functions in the time variable and prove partial Holder continuity of a weak solution near the plane part of the lateral surface of the cylinder. Holder continuity of weak solutions to the correspondent linear problem is stated. A modification of the A(t)-caloric approximation method is applied to study the regularity of weak solutions.

On An Elliptic Equation Arising from Photo-Acoustic Imaging in Inhomogeneous Media

We study an elliptic equation with measurable coefficients arising from photo-acoustic imaging in inhomogeneous media. We establish Holder continuity of weak solutions and obtain pointwise bounds for Green's functions subject to Dirichlet or Neumann condition.

On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems

This paper studies Holder continuity of weak solutions to strongly coupled elliptic systems. We do not assume that the solutions are bounded but BMO and the ellipticity constants can be unbounded.

A regularity theorem for quasilinear parabolic systems under random perturbations

Several aspects of regularity theory for parabolic systems are investigated under the effect of random perturbations. The deterministic theory, when strict parabolicity is assumed, presents both classes of systems where all weak solutions are in fact more regular, and examples of systems with weak solutions which develop singularities in finite time. Our main result is the extension of a regularity result due to Kalita to the stochastic case, which concerns local Holder continuity of weak solutions in the vectorial case. For the proof, we apply stochastic versions of methods, which are classical in the deterministic case (such as difference quotient techniques, higher integrability by embedding theorems, and a version of Moser’s iteration technique). This might be of interest on their own.

Interior W 1,p Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients

We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain \({\Omega \subset \mathbb{R}^{N+1}}\), where the coefficients matrix (aij (z)) is symmetric uniformly positive definite on \({\mathbb{R}^{p_0} (1 \leq p_0 < N)}\). We obtain interior W1,p (1 < p < ∞) regularity and Holder continuity of weak solutions to the equation under the assumption that coefficients aij (z) belong to the \({VMO_L\cap L^\infty}\) and \({({b_{ij}})_{N \times N}}\) is a constant matrix such that the frozen operator \({L_{z_0}}\) is hypoelliptic.

Global strong solvability of a quasilinear subdiffusion problem

We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the modeling of dynamic processes in materials with memory. The proof relies heavily on a regularity result on the interior Holder continuity of weak solutions to time fractional diffusion equations, which has been proved recently by the author. We further establish an L2 decay estimate for the special case with vanishing external source term and homogeneous Dirichlet boundary condition.

Local Hoelder continuity for doubly nonlinear parabolic equations

We give a proof of the Holder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincare inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.

Boundary continuity of solutions to elliptic equations with nonstandard growth

We study regularity properties of weak solutions to elliptic equations involving variable growth exponents. We prove the sufficiency of a Wiener type criterion for the regularity of boundary points. This criterion is formulated in terms of the natural capacity involving the variable growth exponent. We also prove the Holder continuity of weak solutions up to the boundary in domains with uniformly fat complements, provided that the boundary values are Holder continuous.

Lp ESTIMATES FOR WEAK SOLUTIONS TO NONLINEAR SUB-ELLIPTIC SYSTEMS RELATED TO HÖRMANDER’S VECTOR FIELDS

This paper is concerned with the regularity of weak solutions to the general nonlinear sub-elliptic systems related to Hormander’s vector fields. Based on the reverse Holder inequality established by Gianazza, the Lp estimates for weak solutions to the systems under the super-quadratic controllable growth condition and the natural growth condition are established, respectively. Then we obtain the Holder continuity for weak solutions.

Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth

In this paper we are concerned with the regularity in Morrey spaces for weak solutions of a class of degenerate elliptic equations when the coefficient matrices satisfy certain VMO conditions in x uniformly with respect to u and the lower order terms satisfy a natural growth condition. Interior Hölder continuity of weak solutions is also derived with the improvement of the given data regularities.

Hölder Continuity of Weak Solutions for Parabolic Equations with Nonstandard Growth Conditions

In this paper, we investigate the interior regularity including the local boundedness and the interior Holder continuity of weak solutions for parabolic equations of the p(x, t)–Laplacian type. We improve the Moser iteration technique and generalize the known results for the elliptic problem to the corresponding parabolic problem.

On the Hölder Continuity of Solutions of a Certain System Related to Maxwell's Equations

We study the system curl (a(x) curl u )=0 , divu = 0 with a bounded measurable coefficient a(x). The main result of this paper is the Holder continuity of weak solutions of the system above. As an application, we prove the Cα regularity of weak solutions of the Maxwell's equations in a quasi-stationary electromagnetic field.

Liouville Theorems, Partial Regularity and Holder Continuity of Weak Solutions to Quasilinear Elliptic Systems

Liouville Theorems, Partial Regularity and Holder Continuity of Weak Solutions to Quasilinear Elliptic Systems