A Note on Three Classes of Nonlinear Parabolic Equations with Measurable Coefficients

In this note a note and simple technique is presented to replace the complicated one in [1] to obtain Hölder continuity of the weak solutions for a class of nonlinear parabolic equations with measurable coefficients, whose prototype is the singular p-Laplacian. This new approach is also applied to two other classes of nonlinear parabolic equations with measurable coefficients, whose weak solutions exhibit the similar property to those of equations mentioned above.     

Parabolic Q-minima and Their Application

In this paper, the notion named parabolic Q-minima is endowed with rich meanings and its local behavior is investigated. As its direct application we obtain the local regularity, such as boundcdncss, continuity, llolder continuity of the weak solutions of the various filtration equations, e.g., the equation of Newtonian polytropic filtration, the general equation of Newtonian filtration, the equation of elastic filtration, the equation of non-Newtonian polytropic filtration. Therefore, a unifying approach to various regularity results for a great number of nonlinear degenerate parabolic equations is provided.     

A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields

In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result, i.e., the local Hölder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma, the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space. As a consequence, we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.     

The Boundary Regularity of Pseudo-holomorphic Disks

In this paper we will prove the continuity, the C^k-regularity after deforming suitably the domain, and the Hölder continuity, of the weakly pseudo-holomorphic disk with its boundary in a singular totally-real subvariety with only corners as its singularites.     

On the Local Regularity of Solutions for Double Degenerate Nonlinear Parabolic Equations (uq-1)t=div(|ΔU|p-2ΔU) When 1

In this paper, we establish interior Hölder estimates of solutions for double degenerate nonlinear parabolic equations (u^{q-1})_t = div (|∇u|^{p-2}∇u) when 1 < p < 2, p ≤ q.     

A Note on C1,α Estimates for Solutions of Fully Nonlinear Elliptic Equations and Obstacle Problems

We deal with C^{1,α} interior estimates for solutions of fully nonlinear equation F(D²u, Du, x) = f(x) with the bounded gradient Du and a bounded f(x). Based on these estimates we obtain the existence of strong solutions of the obstacle problem for fully nonlinear elliptic equations under natural structure conditions.     

Holder Zygmund Space Techniques to the Navier-Stokes Equations in the Whole Spaces

With the use of Hölder Zygmund space techniques, local regular solutions to the Navier-Stokes equations in R^n are shown to exist when the initial data are in the space {a|(-Δ)^{-β/2}a ∈ C^0(R^n)^n}\quad(0 < β < 1)     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

Local Hardy Spaces and Inhomogeneous Dirichlet Problems in Exterior Regular Domains

In this paper, firstly we give an atomic decomposition of the local Hardy spaces h^p_r(Ω)(O < p ≤ 1) and their dual spaces. where the domain Ω is exterior regular in R^n(n ≥ 3). Then for given data f ∈ h^p_r(Ω), we discuss the inhomogeneous Dirichlet problems: {Lu = f \quad in Ω u = 0 \qquad on ∂ Ω where the operator L is uniformly elliptic. Also we obtain the estimation of Green potential in the local Hardy spaces h^p_r(Ω).     

On W1,q Estimate for Weak Solutions to a Class of Divergence Elliptic Equations

Local W^{1,q} estimates for weak solutions to a class of equations in divergence form D_i(a_{ij}(x)|Du|^{p-2D_ju) = 0 are obtained, where q > p is given. These estimates are very important in obtaining higher regularity for the weak solutions to elliptic equations.     

Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group

In this paper, we concern the divergence Kohn-Laplace equation$$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left( {X_j^*({a^{ij}}{X_i}u) + Y_j^*({b^{ij}}{Y_i}u)} \right)} } + Tu = f - \sum\limits_{i = 1}^n {\left( {X_i^*{f^i} + Y_i^*{g^i}} \right)}$$ with bounded coefficients on the Heisenberg group ${{\mathbb{H}}^n}$, where ${X_1}, \cdots, {X_n},{Y_1}, \cdots, {Y_n}$ and $T$ are real smooth vector fields defined in a bounded region $\Omega \subset {\mathbb{H}^n}$. The local maximum principle of weak solutions to the equation is established. The oscillation properties of the weak solutions are studied and then the Hölder regularity and weak Harnack inequality of the weak solutions are proved.     

Brown运动增量在H?lder范数下的局部泛函Chung重对数律

本文利用Brown运动在H?lder范数下的大偏差和小偏差,得到了Brown运动增量在H?lder范数下的局部泛函Chung重对数律.     

Hölder Continuity to Hamilton-Jacobi Equations with Superquadratic Growth in the Gradient and Unbounded Right-hand Side

We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally 𝒞^{0, α}. Unlike the existing (and more involved) proofs for equations with bounded right-hand side, our arguments rely on constructions of sub- and supersolutions combined with improvement of oscillation techniques.     

Random Fields with Multifractional Regularity Order on Heterogenous Fractal Domains

In this article, we study the effect of the geometry of a domain with variable local dimension on the regularity/singularity of the restriction of a multifractional random field on such a domain. The theories of reproducing kernel Hilbert spaces (RKHS) and generalized random fields are applied. Fractional Sobolev spaces of variable order are considered as RKHSs of random fields satisfying certain elliptic multifractional pseudodifferential equations. The multifractal spectra of these random fields are trivial due to the regularity assumptions on the variable order of the fractional derivatives. In this article, we introduce a family of RKHSs defined by isomorphic identification with the trace on a compact heterogeneous fractal domain of a fractional Sobolev space of variable order. The local regularity/singularity order of functions in these spaces, which depends on the variable order of the fractional Sobolev space considered and on the local dimension of the domain, is derived. We also study the spectral properties of the family of models introduced in the mean-square sense. In the Gaussian case, random fields with sample paths having multifractional local Hölder exponent are covered in this framework.     

Hölder Continuity for Solution Mappings of Parametric Non-convex Strong Generalized Ky Fan Inequalities

Abstract

This article focuses on a new approach to investigate the Hölder continuity for the solution mapping of a parametric non-convex strong generalized Ky Fan inequality. Based on a non-convex separation theorem, the union relation between the solution set of the parametric non-convex strong generalized Ky Fan inequality and the solution sets of a series of Ky Fan inequalities, is established. Without density results and any information on the solution mapping, a sufficient condition for the Hölder continuity of the solution mapping to the parametric non-convex strong generalized Ky Fan inequality is given by using the key union relation. Our method does not impose any convexity, monotonicity, and the single-valuedness of the solution mapping.     

Linear elliptic problems with non-<Emphasis Type="Italic">H</Emphasis><Superscript>−1</Superscript> data and pathological solutions

In the theory of linear elliptic problems with data not belonging to H ^?1 two cases can be distinguished. When the right hand side in the equation is a summable function we point out that the a priori estimates can be attained very quickly by symmetrization methods. On the other side, when the datum includes a distributional term, different and subtler tools have to be used. We deal with an equation in the plane, whose right hand side is a functional on a space of Hölder continuous functions with a suitable exponent. We obtain a priori bounds via duality arguments; these, in addition, show Serrin pathological solution (see Serrin in Ann. Scuola Norm. Sup. Pisa 18(3), 385–387, 1964) in its true light.     

Existence for a model arising from the in situ vitrification process

Existence of a solution is established for a time-dependent problem that can be used to model the in situ vitrification process. Certain properties of the solution are also presented.     

Regularity for Certain Nonlinear Parabolic Systems

Abstract

By means of an inequality of Poincaré type, a weak Harnack inequality for the gradient of a solution and an integral inequality of Campanato type, it is shown that a solution to certain degenerate parabolic system is locally Hölder continuous. The system is a generalization of p-Laplacian system. Using a difference quotient method and Moser type iteration it is then proved that the gradient of a solution is locally bounded. Finally using the iteration and scaling it is shown that the gradient of the solution satisfies a Campanato type integral inequality and is locally Hölder continuous.     

Hölder Continuity of the Saddle Point Set for Real-Valued Functions

This paper is concerned with Hölder continuity of the solution to a saddle point problem. Some new su?cient conditions for the uniqueness and Hölder continuity of the solution for a perturbed saddle point problem are established. Applications of the result on Hölder continuity of the solution for perturbed constrained optimization problems are presented under mild conditions. Examples are given to illustrate the obtained results.     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.