We consider divergence form, second-order strongly parabolic systems in a cylindrical domain with a finite number of subdomains under the assumption that the interfacial boundaries are $C^{1,\text{Dini}}$ and $C^{γ_{0}}$ in the spatial variables and the time variable, respectively. Gradient estimates and piecewise $C^{1/2,1}$-regularity are established when the leading coefficients and data are assumed to be of piecewise Dini mean oscillation or piecewise Hölder continuous. Our results improve the previous results in \cite{ll,fknn} to a large extent. We also prove a global weak type-$(1,1)$ estimate with respect to $A_{1}$ Muckenhoupt weights for the parabolic systems with leading coefficients which satisfy a stronger assumption. As a byproduct, we give a proof of optimal regularity of weak solutions to parabolic transmission problems with $C^{1,μ}$ or $C^{1,\text{Dini}}$ interfaces. This gives an extension of a recent result in \cite{css} to parabolic systems.
We prove a local energy identity for a class of distributional solutions, in $L_{2,\infty} \cap W^{1,0}_2$, of parabolic equations with divergence-free drift.
In this article we establish the exact growth of the solution to the singular quasilinear p-parabolic free boundary problem in non-divergence form near the free boundary from which follows its porosity.
We establish two-sided Gaussian bounds for fundamental solutions of general non-divergence form parabolic operators with Hölder continuous coefficients. The result we obtain is essentially based on parametrix method.
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
We consider an oblique derivative problem in a wedge for nondivergence parabolic equations with discontinuous in $t$ coefficients. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces.
We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C^1 boundary, the other stated for sets with finite perimeter.
We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the $p$-Laplacian type and in non-divergence form. We provide local Hölder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Hölder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.
We establish pointwise estimates for the Green function to the Dirichlet problem for parabolic equation with coefficients measurable in time variable. Using these estimate we obtain coercive estimates for this problem in anisotropic weighted Lebesgue spaces and prove the solvability theorems.
We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces. From this, elliptic and parabolic regularity results are deduced by means of Sneiberg's isomorphism theorem.
We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted $L^p$-estimates for the gradients of weak solutions. Our results improve the classical one to the borderline case by replacing the $L^\infty$-assumption on solutions by solutions in the John-Nirenberg \textup{BMO} space. The results are also generalized to parabolic equations in divergence form with small oscillation elliptic symmetric coefficients and therefore imp…
In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations with respect to the other variables. The corresponding results for equations on a half space are also derived.
We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, Harnack's inequality, Liouville's theorem. The answers are given in terms of the Lebesgue spaces and the Morrey spaces.
We establish the solvability of second order divergence type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be only measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable the coefficients have locally small mean oscillations. We also obtain the corresponding $W^1_p$-solvability of second order elliptic systems in divergence form. Our results are new even for scalar equations and the proofs simplify the methods used previously in [12]
In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hormander's vector fields and establish a Nash type result, i.e., the local Holder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of Hormander's vector fields and a De Giorgi type Lemma, the Holder 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 Holder space. As a consequence, we give the Ha…
We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted $L_{p,q}$-spaces. The concept of "critical exponent" introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi-linear non-divergence parabolic equations with discontinuous in time coefficients in cylinders $Ω\times(0,T)$, where $Ω$ is a bounded domain with an edge or with a conical point.
We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for {\em stochastic} partial differential equations with BMO$_x$ coefficients.
We consider an oblique derivative problem for non-divergence parabolic equations with discontinuous in $t$ coefficients in a half-space. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces. We also give an application of this result to linear parabolic equations in a bounded domain. In particular, if the boundary is of class ${\cal C}^{1,δ}$, $δ\in (0,1]$, then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary.
In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations $$\partial_t u-|Du|^γΔ_p^N u=f,$$ where $-1<γ<0$, $1<p<\infty$, and $f$ is a given bounded function. We establish interior Hölder regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.
In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic counterpart of the equations studied by Julin in [10] where a generalized Harnack inequality was obtained which quantifies the strong maximum principle. Our version of parabolic Harnack (see Theorem 1.2) when restricted to the elliptic case is however quite different from that in [10]. The key new feature of this work is an appropriate modification of the sta…
We prove Schauder estimates for solutions to both divergence and non-divergence type higher-order parabolic systems in the whole space and the half space. We also provide an existence result for divergence type systems in a cylindrical domain. All coefficients are assumed to be only measurable in the time variable and Hölder continuous in the spatial variables.
We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift \[-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)\] in some domain $Q\subset \mathbb{R}^{n+1}$. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift $b$ is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is…
We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in $(-\infty, T) \times \mathbb{R}^d_+$, where $\mathbb{R}^d_+ = \{x \in \mathbb{R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given, and the diffusion matrices are the product of $x_d$ and bounded uniformly elliptic matrices, which are degenerate at $\{x_d=0\}$. As such, our class of equations resembles well the corresponding class of degenerate viscous Hamilton-Jacobi equations. We obtain wellposedness results and regularity…
In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=νΔ+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;ξ,τ)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension…
We investigate the parabolic Boundary Harnack Principle for both divergence and non-divergence type operators by the analytical methods we developed in the elliptic context. Besides the classical case, we deal with less regular space-time domains, including slit domains.
We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift \[-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)\] in some domain $Ω\subset \mathbb{R}^{n+1}$. We prove a variant of Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate with $L^{p}$ norm of the inhomogeneous term for some number $p<n+1$. Based on it, we derive the growth theorems and the interior Harnack inequality. In this paper, we will only assume the drift $b$ is in certain Morrey spaces defined below which are critical under the parabolic scaling but not…
Let $\mathrm{X}=(X_{1},...,X_{q})$ be a family of real smooth vector fields satisfying Hömander's condition. The purpose of this paper is to establish gradient estimates in generalized Morrey spaces for weak solutions of the divergence degenerate parabolic system related to $X$ :%\[u_{t}^{i}+X_α^{\ast}(a_{ij}^{αβ}(z)X_βu^{j}%)=g_{i}+X_α^{\ast}f_{i}^α(z), \] where $α,β=1,2,...,q,$ $i,j=1,2,...,N$, $X_α^{\ast}$ is the transposed vector field of $X_α$, $z=(t,x)\in{\mathbb{R}}^{n+1}$, and coefficients $a_{ij}^{αβ}(z)$ belong to the space $VMO$ induced by the vector fields $X_{1}, ...,X_{q}$.
An $L_{p}$-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all functions independent of $x$. Weak uniqueness of the martingale problem associated with such equations is obtained.
In this paper we study the fundamental solution $\varGamma(t,x;τ,ξ)$ of the parabolic operator $L_{t}=\partial_{t}-Δ+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case that $b$ belongs to the Lebesgue space $L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regularity of weak solutions to the parabolic equation $L_{t}u=0$ depends critically on the value of the parabolic exponent $γ=\frac{2}{l}+\frac{n}{q}$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $γ\leq1$, …
We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Hölder semi-norms not with respect to all, but only with respect to some of the independent variables.
James Serrin's fundamental contributions to the theory of quasilinear elliptic equations are well-known and widely appreciated. He also made less well-known contributions to the theory of quasilinear parabolic equations which we dicuss in this note. Jurgen Moser gave greatly simplified proofs of the De Giorgi-Nash regularity results for linear divergence structure elliptic and parabolic differential equations using an original iterative technique. Serrin extended Moser's techniques and applied them to the study of divergence structure quasilinear elliptic equations, and in collaboration with A…
The properties of the solution for the Cauchy problem of a double nonlinear time-dependent parabolic equation in non-divergence form with a source or absorption
We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb R^d_+ = \{x \in \mathbb R^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $μ(x_d)$ and bounded uniformly elliptic matrices, where $μ(x_d)$ behaves like $x_d^α$ for some given $α\in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. Under a partially VMO assumption on th…
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition.
An $L_{q}(L_{p})$-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all measurable functions depending only on $t$. The method of proving simplifies the methods previously used in the case $p=q$.
In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO regularity conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed case. For the proof, we establish both interior and boundary Lipschitz estimates f…
We study the regularity of the solution of the double obstacle problem form for fully non linear parabolic and elliptic operators. We show that when the obstacles are sufficiently regular the solution is $C^{1,α}$ in the interior for both the parabolic and the elliptic cases.
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
We study the generation property for a fourth order operator in divergence or in non divergence form with suitable Neumann boundary conditions. As a consequence we obtain the well posedness for the parabolic equations governed by these operators. The novelty of this paper is that the operators depend on a function $a: [0,1] \rightarrow \R_+$ that degenerates somewhere in the interval.
Under various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Hölder semi-norms not with respect to all, but only with respect to some of the independent variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.
In this paper we continue the study initiated in [FGN] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X = {X_1,...,X_q} in R^n with C^1-coefficients satisfying Hörmander's finite rank condition, i.e., the rank of Lie{X_1,...,X_q} equals n at every point in R^n. In [FGN] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at t…
This work focuses on drift-diffusion equations with fractional dissipation $(-Δ)^α$ in the regime $α\in (1/2,1)$. 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 $β\in (0,1)$, the $C^β$ norm of the solution depends only on the size of the drift in critical spaces of the form $L^{q}_{t}(BMO^{-γ}_{x})$ with $q>2$ and $γ\in (0,2α-1]$, along with the $L^{2}_{x}$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.
In this note we define and study the stochastic process $X$ in link with a parabolic transmission operator $(A,D(A))$ in divergence form. The transmission operator involves a diffraction condition along a transmission boundary. To that aim we gather and clarify some results coming from the theory of Dirichlet forms as exposed in [6] and [14] for general divergence form operators. We show that $X$ is a semimartingale and that it is solution of a stochastic differential equation involving partial reflections in the co-normal directions along the transmission boundary.
