Hölder estimates for quasilinear doubly degenerate parabolic equations

For positive bounded generalized solutions of degenerate parabolic equations of the form t 0, m 2, one establishes local Hölder estimates, independent of the lower bounds of the indicated solutions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 171, pp. 70–105, 1989.     

Hölder estimate for the solution of degenerate parabolic equations

Consider the degenerate parabolic equations of the type $$u_t = div A(x,t,u,Du) + b(x,t,u,Du)$$ which is of the same nature as $$u_t = div|Du|^p Du + |Du|^{p + 2} (p > 2).$$ This paper is to study the \(C^{1 + \alpha } ,\frac{{1 + \alpha }}{2}\) Hölder continuity of a class of degenerate parabolic equations and the existence and uniqueness of the initial boundary value problem.     

Hölder regularity for nondivergence nonlocal parabolic equations

This paper proves Hölder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first proves Hölder continuity for viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative.     

Local Hölder continuity of nonnegative weak solutions of degenerate parabolic equations

In this paper, we consider nonnegative weak solutions for a class of degenerate parabolic equations. Using Moser’s method, we get the local boundedness of solutions to equations of this class. Then we prove that the solutions are locally Hölder continuous.     

Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations

A two-parameter family of Harnack type inequalities for non-negative solutions of a class of singular, quasilinear, homogeneous parabolic equations is established, and it is shown that such estimates imply the Hölder continuity of solutions. These classes of singular equations include p-Laplacean type equations in the sub-critical range ${1 < p \le\frac{2N}{N+1}}$ and equations of the porous medium type in the sub-critical range ${0 < m \le\frac{(N-2)_+}{N}}$ .     

Interior Hölder and gradient estimates for the homogenization of the linear elliptic equations

Hölder and gradient estimates for the correctors in the homogenization are presented based on the translation invariance and Li-Vogelius’s gradient estimate. If the coefficients are piecewise smooth and the homogenized solution is smooth enough, the interior error of the first-order expansion is O(?) in the Hölder norm; it is O(?) in W ^1,∞ based on the Avellaneda-Lin’s gradient estimate when the coefficients are Lipschitz continuous. These estimates can be partly extended to the nonlinear parabolic equations.     

H?lder estimates for elliptic equations degenerate on part of the boundary of a domain

The present paper studies the Dirichlet problem for elliptic equations degenerate on part of the boundary of a domain and the degeneracy is of the Keldysh type. By introducing a proper metric that is related to the operator we establish the global H?lder estimates when some well-posed boundary conditions are satisfied. The main methods are the construction of some barrier functions and the interpolation of the estimates of uniformly elliptic operators.     

Hölder estimates of solutions to difference partial differential equations of elliptic-parabolic type

For solutions to difference partial differential equations of elliptic-parabolic type, there is achieved Hölder estimates independent of the time discrete mesh.     

Hölder continuity of solutions to quasilinear elliptic equations involving measures

We show that the solutionu of the equation

Hölder continuity of solutions to a degenerate elliptic equation in the presence of the Lavrentiev phenomenon

We study a second-order elliptic equation for which the Dirichlet problem can be posed in a nonunique way due to the so-called Lavrentiev phenomenon. In the corresponding weighted Sobolev space smooth functions are not dense, which leads to the existence of W – solutions and H – solutions. For H - solutions, we establish the Hölder continuity. We also discuss this question for W – solutions, for which the situation is more complicated.     

Global Hölder estimates for equations of Monge-Ampère type

Summary We prove interior oscillation and global Hölder estimates, independent of any boundary data, for convex solutions of certain types of Monge-Ampère equations under suitable conditions on the equation and the domain ⁿ . We also deduce the existence, uniqueness, regularity and unboundedness, under suitable conditions, of convex extremal solutions of certain Monge-Ampère equations.     

Solutions in Hölder spaces of boundary-value problems for parabolic equations with nonconjugate initial and boundary data

The first and second one-dimensional boundary-value problems for parabolic equations are investigated in the case where the conjugation conditions for all required orders are not satisfied. The existence and uniqueness are proved. Estimates of solutions in classical and weighted Hölder spaces are obtained. We prove that the violation of conjugation for the given functions on the boundary of the domain at the initial-time moment causes the appearance of singular solutions. The order of singularity (as a power of t) is found for the singular solutions for t = 0.     

Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation ${(?\mathrm{\Delta })}^{\alpha }$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta }$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q({\mathrm{BMO}}_x^{?\gamma })$ with $q>2$ and $\gamma \in (0,2\alpha ?1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.