Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces

We consider the first boundary value problem and the oblique derivative problem for a linear second-order parabolic equation in noncylindrical not necessarily bounded domains with nonsmooth (with respect to t) and noncompact lateral boundary under the assumption that the right-hand side and the lower-order coefficients of the equation may have certain growth when approaching the parabolic boundary of the domain and all coefficients are locally Hölder with given characteristics of the Hölder property. We construct a smoothness scale of solutions of these boundary value problems in Hölder spaces of functions that admit growth of higher derivatives near the parabolic boundary of the domain.     

Regularity of solutions of boundary value problems for parabolic equations of arbitrary order in weighted Hölder spaces

We consider initial-boundary value problems for a uniformly parabolic equation of arbitrary order 2m in a noncylindrical domain whose lateral boundary is nonsmooth with respect to t. We assume that the lower-order coefficients and the right-hand side of the equation, generally speaking, grow to infinity no more rapidly than some power function when approaching the parabolic boundary of the domain, all coefficients of the equation are locally Hölder, and their Hölder constants can grow near that boundary. We construct a smoothness scale of solutions of such problems in weighted Hölder classes of functions whose higher derivatives may grow when approaching the parabolic boundary of the domain.     

Model oblique derivative problem for the heat equation in the Zygmund space <Emphasis Type="Italic">H</Emphasis> <Subscript>1</Subscript>

The third boundary value problem and the oblique derivative problem for the heat equation are considered in model formulations. A difference compatibility condition is introduced for the initial and boundary functions. Under suitable assumptions made about the problem data, the solutions are shown to belong to the parabolic Zygmund space H₁, which is the analogue of the parabolic Hölder space for an integer smoothness exponent.     

Regularity of the solution of the Cauchy problem for a higher-order parabolic equation

We prove the unique solvability of the Cauchy problem in a weighted Hölder space for a linear parabolic equation of order 2m under the condition that the lower coefficients and the right-hand side of the equation can have certain growth when approaching the plane that is the support of the initial data, while the higher coefficients do not necessarily satisfy the Dini condition near this plane.We construct a smoothness scale of solutions of the Cauchy problem in the corresponding weighted Hölder spaces.     

Backward Euler Scheme,Singular Hölder Norms,and Maximal Regularity for Parabolic Difference Equations

We show finite difference analogues of maximal regularity results for discretizations of abstract linear parabolic problems. The involved spaces are discrete versions of spaces of Hölder continuous functions, which can be singular in 0. The main tools are real interpolation and Da Prato–Grisvard's theory of the sum of linear operators.     

Uniqueness of Solution to the First Initial Boundary Value Problem for Parabolic Systems on the Plane in a Model Case

The first initial boundary value problem for a one-dimensional (in x) Petrovskii parabolic second-order system with constant coefficients in a semibounded (in x) domain with a nonsmooth lateral boundary is proved to have a unique classical solution in certain Hölder classes.     

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate the Hölder regularity of solutions to the time fractional Schrödinger equation of order 1<α<2, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in Hölder spaces. In addition, we also prove Hölder regularity result for the Schrödinger equation.     

Linear two-phase Venttsel problems

A priori estimates are established for the two-phase boundary value problems with Venttsel interface conditions for linear nondivergent parabolic and elliptic equations. By these estimates, the existence and uniqueness theorems in Sobolev and Hölder spaces are proved.     

Time regularity of the densities for the Navier–Stokes equations with noise

We prove that the density of the law of any finite-dimensional projection of solutions of the Navier–Stokes equations with noise in dimension three is Hölder continuous in time with values in the natural space L ¹. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected, at least with the known regularity, diffusive scaling.     

Dirichlet problem for parabolic systems with Dini continuous coefficients on the plane

The Dirichlet problem for a one-dimensional (with respect to x) second-order parabolic system with Dini continuous coefficients is considered in an x-semibounded domain with a nonsmooth lateral boundary from the Dini–Hölder class. The classical solvability of the problem is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Solvability of a Model Oblique Derivative Problem for the Heat Equation in the Zygmund Space <Emphasis Type="Italic">H</Emphasis><Subscript>1</Subscript>

We consider the oblique derivative problem for the heat equation in a model statement. We introduce a difference matching condition for the initial and boundary functions, under which we establish conditions on the data of the problem sufficient for the solution to belong to the parabolic Zygmund space H₁, which is an analog of the parabolic Hölder space for the case of an integer smoothness exponent. We present an example showing that if the above-mentioned matching condition is not satisfied, then the solution may fail to belong to the space H₁.     

Solutions of model heat problems in Zygmund spaces

We consider the first boundary value problem and the oblique derivative problem for the heat equation in the model case where the domain is a half-layer and the coefficients of the boundary operator in the oblique derivative problem are constant. Under the corresponding assumptions on the problem data, we show that the solutions belong to anisotropic Zygmund spaces, which “close” the scale of anisotropic Hölder spaces for integer values of the smoothness exponent.     

Positive Solutions in the Space of Lipschitz Functions for Fractional Boundary Value Problems with Integral Boundary Conditions

In this paper, we study the existence of positive solutions for a class of nonlinear fractional boundary value problems with integral boundary conditions in Hölder spaces. Our analysis relies on a sufficient condition for the relative compactness in Hölder spaces and the classical Schauder fixed point theorem.     

A Free Boundary Problem of Fourth Order: Classical Solutions in Weighted Hölder Spaces

We prove short-time existence and uniqueness of classical solutions in weighted Hölder spaces for the thin-film equation with linear mobility, zero contact angle, and compactly supported initial data. We furthermore show regularity of the free boundary and optimal regularity of the solution in terms of the regularity of the initial data. Our approach relies on Schauder estimates for the operator linearized at the free boundary, obtained through a variant of Safonov's method that is solely based on energy estimates.     

Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups,with Applications to Earth Mover’s Distance

We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Hölder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Hölder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.     

Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions

We consider a system of three surfaces, graphs over a bounded domain in ?², intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to 2π/3.) For the corresponding two-dimensional parabolic free boundary problem we prove short-time existence of classical solutions (in parabolic Hölder spaces), for sufficiently regular initial data satisfying a compatibility condition.     

Sharp Regularity for Elliptic Systems Associated with Transmission Problems

The paper concerns regularity theory for linear elliptic systems with divergence form arising from transmission problems. Estimates in BMO, Dini and Hölder spaces are derived simultaneously and the gaps among of them are filled by using Campanato–John–Nirenberg spaces. Results obtained in the paper are parallel to the classical regularity theory for elliptic systems.     

Classical solvability for one-dimensional, free boundary, Florin, Muskat-Verigin, and Stefan problems

Three one-dimensional problems with free boundary for second order parabolic equations are considered, namely, the Florin, Muskat-Verigin, and Stefan problems. Existence and uniqueness theorems in the weighted Hölder spaces are established for these problems. Coercive estimates for solutions are found. Bibliography:44 titles.     

Generalized Hölder Spaces of Holomorphic Functions in Domains in the Complex Plane

We study some nonstandard spaces of functions holomorphic in domains on the complex plain with certain smoothness conditions up to the boundary. The first type is the space of Hölder-type holomorphic functions with prescribed modulus of continuity \(\omega =\omega (h)\), and the second is the variable exponent holomorphic Hölder space with the modulus of continuity \(|h|^{\lambda (z)}\). We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.     

Hölder estimates for solutions of the Cauchy problem for the porous medium equation with external forces

We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. The quantitative estimates are important for studying the perturbation problem of the porous medium equation. We obtain the scale invariant Hölder estimates for weak solutions of the porous medium equations with the external forces. As a particular case, we recover the well known Hölder estimates for the linear heat equation.