Interior Regularity Research Articles

On higher dimensional complex Plateau problem

Let X be a compact connected strongly pseudoconvex CR manifold of real dimension $$2n-1$$ in $${\mathbb {C}}^{N}$$ . It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $$n\ge 3$$ and $$N=n+1$$ , Yau found a necessary and sufficient condition for the interior regularity of the Harvey–Lawson solution to the complex Plateau problem by means of Kohn–Rossi cohomology groups on X in 1981. For $$n=2$$ and $$N\ge n+1$$ , the first and third authors introduced a new CR invariant $$g^{(1,1)}(X)$$ of X. The vanishing of this invariant will give the interior regularity of the Harvey–Lawson solution up to normalization. For $$n\ge 3$$ and $$N>n+1$$ , the problem still remains open. In this paper, we generalize the invariant $$g^{(1,1)}(X)$$ to higher dimension as $$g^{(\Lambda ^n 1)}(X)$$ and show that if $$g^{(\Lambda ^n 1)}(X)=0$$ , then the interior has at most finite number of rational singularities. In particular, if X is Calabi–Yau of real dimension 5, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization.

Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian equation

We consider the pseudo-$p$-Laplacian operator: $$ \tilde \Delta_p u = \sum_{i=1}^N \partial_i (|\partial_i u|^{p-2} \partial_i u)=(p-1) \sum_{i=1}^N |\partial_i u|^{p-2} \partial_{ii} u \ \text{ for $p > 2$.} $$ We prove interior regularity results for the viscosity (resp. weak) solutions in the unit ball $B_1$ of $\tilde \Delta_p u =(p-1) f$ for $f\in { \mathcal C} (\overline{B_1})$ (resp. $f\in L^\infty(B_1)$). First, the Hölder local regularity for any exponent $\gamma < 1$, recovering in that way a known result about weak solutions. Second, we prove the Lipschitz local regularity.

Dirichlet control of elliptic state constrained problems

We study a state constrained Dirichlet optimal control problem and derive a priori error estimates for its finite element discretization. Additional control constraints may or may not be included in the formulation. The pointwise state constraints are prescribed in the interior of a convex polygonal domain. We obtain a priori error estimates for the $$L^2(\varGamma )$$L2(Γ)-norm of order $$h^{1-1/p}$$h1-1/p for pure state constraints and $$h^{3/4-1/(2p)}$$h3/4-1/(2p) when additional control constraints are present. Here, p is a real number that depends on the largest interior angle of the domain. Unlike in e.g. distributed or Neumann control problems, the state functions associated with $$L^2$$L2-Dirichlet control have very low regularity, i.e. they are elements of $$H^{1/2}(\varOmega )$$H1/2(Ω). By considering the state constraints in the interior we make use of higher interior regularity and separate the regularity limiting influences of the boundary on the one-hand, and the measure in the right-hand-side of the adjoint equation associated with the state constraints on the other hand. We note in passing that in case of control constraints, these may be interpreted as state constraints on the boundary.

Interior Regularity for Regional Fractional Laplacian

In this paper, we study interior regularity properties for the regional fractional Laplacian operator. We obtain the integer order differentiability of the regional fractional Laplacian, which solves a conjecture of Guan and Ma (Probab. Theory Related Fields 134:649–694, 2006). We further extend the integer order differentiability to the fractional order of the regional fractional Laplacian. Schauder estimates for the regional fractional Laplacian are also provided.

A sign-changing Liouville equation

We examine periodic solutions to an initial boundary value problem for a Liouville equation with sign-changing weight. A representation formula is derived for both singular and nonsingular boundary data, including data arising from fractional linear maps. In the case of singular boundary data, we study the effects that the induced singularity has on the interior regularity of solutions. Regularity criteria are also found for a generalized form of the equation.

Fractional elliptic equations, Caccioppoli estimates and regularity

Let L=−divx(A(x)∇x) be a uniformly elliptic operator in divergence form in a bounded domain Ω. We consider the fractional nonlocal equations{Lsu=f,inΩ,u=0,on∂Ω,and{Lsu=f,inΩ,∂Au=0,on∂Ω. Here Ls, 0<s<1, is the fractional power of L and ∂Au is the conormal derivative of u with respect to the coefficients A(x). We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients A(x), the right hand side f and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman–Stampacchia–Weinberger and we obtain nonlocal integro-differential formulas for Lsu(x). Essential tools in the analysis are the semigroup language approach and the extension problem.

Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations I: Bellman Equations with Constant Coefficients

We consider a stochastic optimal control problem in a domain, in which the diffusion coefficients, drift coefficients, and discount factor are independent of the spatial variables. Under appropriate assumptions, for k=0,1, when the terminal and running payoffs are globally $C^{k,1}$, we establish the interior $C^{k,1}$-smoothness of the value function, which yields the existence and uniqueness of the locally $C^{k,1}$-solution to the associated Dirichlet problem for the possibly degenerate elliptic Bellman equation with constant coefficients. Estimates for first and second derivatives of the solution are also obtained.

On the regularity of solutions of the Cauchy problem for the Zakharov-Kuznetsov equation in Hölder norms

The problem of the interior regularity of generalized solutions of the Cauchy problem for the Zakharov-Kuznetsov equation is studied. The existence of Holder-continuous derivatives of given solutions is established. The study is based on the properties of the fundamental solution of the corresponding linearized equation.

Higher Integrability of Solutions to Generalized Stokes System Under Perfect Slip Boundary Conditions

We prove an Lq theory result for generalized Stokes system in a \({\mathcal{C}^{2,1}}\) domain complemented with the perfect slip boundary conditions and under Φ-growth conditions. Since the interior regularity was obtained in Diening and Kaplický (Manu Math 141:336–361, 2013), a regularity up to the boundary is an aim of this paper. In order to get the main result, we use Calderon–Zygmund theory and the method developed in Caffarelli and Peral (Ann Math 130:189–213, 1989). We obtain higher integrability of the first gradient of a solution.

Nonlinear gradient estimates for parabolic obstacle problems in non-smooth domains

We study a nonlinear parabolic equation with an irregular obstacle over a non-smooth domain and establish a global Calderon–Zygmund theory by proving that the spatial gradient of the weak solution is as integrable as both that of the obstacle and the nonhomogeneous term. Our results extend the known interior regularity for such obstacle problems in Lebesgue spaces to the boundary one in Orlicz spaces. The domain under consideration is a δ-Reifenberg domain which is a natural extension of a Lipschitz one with a small Lipschitz constant, while the function space under consideration is an Orlicz space which is a natural generalization of the Lebesgue space.

On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

We establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-Sverak’s criterion. We also show that if a weak solution u satisfies $$\left\| {u( \cdot ,t)} \right\|_{L^p } \leqslant C( - t)^{(3 - p)/2p} $$ for some 3 < p < ∞, then the number of singular points is finite.

New monotonicity formulas and the optimal regularity in the Signorini problem with variable coefficients

We study the interior Signorini, or lower-dimensional obstacle problem for a uniformly elliptic divergence form operator L=div(A(x)∇) with Lipschitz continuous coefficients. Our main result states that, similarly to what happens when L=Δ, the variational solution has the optimal interior regularity Cloc1,12(Ω±∪M), when M is a codimension one flat manifold which supports the obstacle. We achieve this by proving some new monotonicity formulas for an appropriate generalization of the celebrated Almgren's frequency functional.

A Perturbation Argument for a Monge–Ampère Type Equation Arising in Optimal Transportation

We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampere equation in order to obtain, first, interior C1,1 estimates for the potential and, second, interior Holder estimates for second derivatives. In particular, we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary.

Optimal A Priori Error Estimates for an Elliptic Problem with Dirac Right-Hand Side

It is well known that finite element solutions for elliptic PDEs with Dirac measures as source terms converge, due to the fact that the solution is not in $H^1$, suboptimal in classical norms. A standard remedy is to use graded meshes, then quasioptimality, i.e., optimal up to a log-factor, for low order finite elements can be recovered, e.g., in the $L^2$-norm. Here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi-uniform meshes in an $L^2$-seminorm. The seminorm is defined as an $L^2$-norm on a fixed subdomain which excludes the locations of the delta source terms. Our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error, e.g., in dimension reduced settings or is not the quantity of interest, e.g., in optimal control problems. The quasi-optimal and optimal order a priori bounds are obtained recursively by using Aubin--Nitsche techniques, local Wahlbin-type error estimates, interior regularity results, and weighted Sobolev norms. For the proof of these results no graded meshes are required, it is sufficient to work on a family of quasi-uniform meshes. Numerical tests in two and three space dimensions confirm our theoretical results.

C 1, α Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels

We prove a C 1, α interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.

Campanato estimates for the generalized Stokes System

We study interior regularity of solutions of a generalized stationary Stokes problem in the plane. The main, elliptic part of the problem is given in the form $$\mathrm{div }(\mathbf{A}(\mathbf{D}\mathbf{u}))$$ , where $$\mathbf{D}$$ is the symmetric part of the gradient. The model case is $$\mathbf{A}(\mathbf{D}\mathbf{u})=(\kappa +{|{\mathbf{D}\mathbf{u}}|})^{p-2}\mathbf{D}\mathbf{u}$$ . We show optimal $$\mathrm{BMO}$$ and Campanato estimates for $$\mathbf{A}(\mathbf{D}\mathbf{u})$$ . Some corollaries for the generalized stationary Navier–Stokes system and for its evolutionary variant are also mentioned.

Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace

We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``post-Carleman estimates proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.

Douglis–Nirenberg elliptic systems in Hörmander spaces

Douglis–Nirenberg systems uniformly elliptic in ℝ n are studied in the class of Hormander Hilbert spaces H φ ; where φ is an RO-varying function of scalar argument. An a priori estimate is established for the solutions, and their interior regularity is investigated. A sufficient condition under which these systems possess the Fredholm property is obtained.

A Probabilistic Approach to Interior Regularity of Fully Nonlinear Degenerate Elliptic Equations in Smooth Domains

We consider the value function of a stochastic optimal control of degenerate diffusion processes in a domain D. We study the smoothness of the value function, under the assumption of the non-degeneracy of the diffusion term along the normal to the boundary and an interior condition weaker than the non-degeneracy of the diffusion term. When the diffusion term, drift term, discount factor, running payoff and terminal payoff are all in the class of $C^{1,1}(\bar{D})$ , the value function turns out to be the unique solution in the class of $C_{loc}^{1,1}(D)\cap C^{0,1}(\bar{D})$ to the associated degenerate Bellman equation with Dirichlet boundary data. Our approach is probabilistic.

On the Interior Regularity Criteria for Suitable Weak Solutions of the Magnetohydrodynamics Equations

We present new interior regularity criteria for suitable weak solutions of the magneto-hydrodynamics (MHD) equations in terms of the velocity field. The result means that the velocity field plays a more important role than the magnetic field in the local regularity theory of the MHD equations. This also gives a positive answer to the problem proposed by Kang and Lee in [J. Differential Equations, 247 (2009), pp. 2310--2330].