In this paper we consider maximal regularity for the vector-valued quasi-steady linear elliptic problems. The equations are the elliptic equation in the domain and the evolution equations on its boundary. We prove the maximal $L_p$-$L_q$ regularity for these problems and give examples that our results are applicable. The Lopatinskii--Shapiro and the asymptotic Lopatinskii--Shapiro conditions are important to get boundedness of solution operators.
In this paper we study the regularity of weak solutions to systems of p-Stokes type, describing the motion of some shear thinning fluids in certain steady regimes. In particular we address the problem of regularity up to the boundary improving previous results especially in terms of the allowed range for the parameter p.
We extend the Caffarelli-Kohn-Nirenberg type partial regularity theory for the steady $5$-dimensional fractional Navier-Stokes equations with external force to the hyperdissipative setting. In our argument we use the methods of Colombo-De Lellis-Massaccesi to apply a blowup procedure adapted from work of Ladyzhenskaya-Seregin.
Under the condition of small external forces, we obtain existence of a weak solution of the steady Hall-MHD system with Hölder continuous magnetic field. We also established regularity of weak solutions provided that magnetic fields are bounded. For sufficiently small external forces, uniqueness result is also established.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $Ω\subset \Bbb R^3$. In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $Ω=\Bbb R^3$, we show that the set of possible singularities is compact.
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
In this note we study the global regularity in the Morrey spaces for the second derivatives for the strong solutions of non variational elliptic equations.
We obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations leaded by operator that are neither convex/concave nor uniformly elliptic.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
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.
Due to degeneracy near the boundary, the question of high regularity for solutions to the steady Prandtl equations has been a longstanding open question since the celebrated work of Olenick. We settle this open question in affirmative in the absence of an external pressure. Our method is based on energy estimates for the quotient, $q = \frac{v}{\bar{u}}$, $\bar{u}$ being the classical Prandtl solution, via the linear Derivative Prandtl Equation (LDP). As a consequence, our regularity result leads to the construction of Prandtl layer expansion up to any order.
We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by Galerkin methods. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.
In this paper, we study the regularity of weak solutions to a class of nonlocal elliptic equations in Bessel potential spaces $H^{s,p}$. Our main results can be seen as an extension of the well-known $W^{1,p}$ regularity theory for local second-order elliptic equations in divergence form to the nonlocal setting.
We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.
Consider steady-state weak solutions to the incompressible Navier-Stokes equations in six spatial dimensions. We prove that the 2D Hausdorff measure of the set of singular points is equal to zero. This problem was mentioned in 1988 by Struwe [24], during his study of the five dimensional case.
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
We establish partial regularity result for vector-valued solutions to second order elliptic system in divergence form. The coefficients safisfies Dini condition respect to $(x,u)$ with growth order lager than 2. We prove $C^1$-regularity of the solutions outside of singular set.
Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev-Triebel-Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce Fredholm bounded operators on appropriate pairs of these spaces.
We prove a Brezis--Kato regularity type results for solutions of the higher order nonlinear elliptic equation \[ L u = g(x,u)\qquad\text{in }Ω\] with an elliptic operator $L$ of $2m$ order with variable coefficients and a Carathéodory function $g:Ω\times \mathbb{C}\to\mathbb{C}$, where $Ω\subset\mathbb{R}^N$ is an open set with $N > 2m$.
It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes and MHD equations are Hölder continuous near boundary provided that either $r^{-3}\int_{B_r^+}|u(x)|^3dx$ or $r^{-2}\int_{B_r^+}|\nabla u(x)|^2dx$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong-Strain \cite{DS}.
We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,α}$ on the boundary of the domain, for an optimal and explicit $α$ given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian;…
We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.
In this paper, we consider the regularity of solutions to elliptic equation with Grushin's operator. By using the Feynman-Kac formula, we first get the expression of heat kernel, and then by using the properties of heat kernel, the optimal regularity of solutions will be obtained. The novelty of this paper is that the Grushin's operator is a degenerate operator.
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.
We establish partial regularity for vector-valued solutions to inhomogeneous elliptic systems in divergence form where the coefficients are possibly discontinuous with respect to $x$. More precisely, we assume a VMO-condition with respect to the $x$ and continuity with respect to $u$ and prove Hölder continuity of the solutions outside of singular sets.
We study the uniqueness and regularity of the steady states of the diffusively driven Boltzmann equation in the physically relevant case where the restitution coefficient depends on the impact velocity including, in particular, the case of viscoelastic hard-spheres. We adopt a strategy which is novel in several aspects, in particular, the study of regularity does not requires a priori knowledge of the time-dependent problem. Furthermore, the uniqueness result is obtained in the small thermalization regime by studying the so-called quasi-elastic limit for the problem. An important new aspect li…
In this note we prove an end-point regularity result on the Lp integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations.
Using a standard linearization technique and previously obtained microlocal properties for pseudodifferential operators with smooth coefficients, the authors state results of microlocal regularity in generalized Besov spaces for solutions to non linear PDE.
In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^α$, $C^{1,α}$ and $C^{2,α}$ regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations.
In this paper we study a class of anisotropic equations with a lower order term whose coefficients lay in Marcinkiewicz spaces. We prove some regularity results for local solutions requiring any control on the norm of the coefficients.
We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.
We prove existence and regularity results for weak solutions of non linear elliptic systems with non variational structure satisfying $(p,q)$-growth conditions. In particular we are able to prove higher differentiability results under a dimension-free gap between $p$ and $q$.
In this work we establish local $C^{2,α}$ regularity estimates for flat solutions to non-convex fully nonlinear elliptic equations provided the coefficients and the source function are of class $C^{0,α}$. For problems with merely continuous data, we prove that flat solutions are locally $C^{1,\text{Log-Lip}}$.
In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.
We consider nonlinear fourth order elliptic equations of double divergence type. We show that for a certain class of equations where the nonlinearity is in the Hessian, solutions that are C^{2,alpha} enjoy interior estimates on all derivatives.
In the present paper, we study sharp C^{1;α} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.
This manuscript aims to provide a self-contained introduction to the regularity theory for elliptic PDE, focusing on the main ideas rather than proving all results in their greatest generality. It can be seen as a bridge between an elementary PDE course and more advanced textbooks. This is a draft of the book "Regularity Theory for Elliptic PDE". The final version has been published in Zurich Lectures in Advanced Mathematics, EMS Press, 2022.
We study geometric and regularity properties of the largest subsolution of a one-phase free boundary problem under a very general free boundary condition in R2. Moreover, we provide density bounds for the positivity set and its complement near the free boundary.
This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.
We prove regularity estimates for weak solutions to the Dirichlet problem for a divergence form elliptic operator. We give $L^p$ estimates for the second derivative for $p<2$. Our work generalizes results due to Miranda [28].
We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for w…
We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-Δ)^s$ on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\textrm{loc}}(Ω)$, while for $2\leq p<\infty$ we show that the solutions belong to $W^{2s,p}_{\textrm{loc}}(Ω)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working dir…
Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradi…
We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients.
