We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$ u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u) $$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivière and the last named author.
We discuss recent advances in the regularity problem of a variety of fluid equations and systems. The purpose is to illustrate the advantage of harmonic analysis techniques in obtaining sharper conditional regularity results when compared to classical energy methods.
In my previous paper I have contrived a Ginzburg-Landau heat flow with a time-dependent parameter and by using it, I constructed a harmonic heat flow into spheres with a monotonical inequality and a reverse Poincaré inequality. This paper establishes these two energy inequalities near the boundary and then by making the best of them, we discuss a partial boundary regularity. In addition to it, we demonstrate a whole domain's regularity under "the one-sided condition." This has been proposed by S.Hildebrandt and K.-O.Widman.
We study the Sobolev regularity of $p$-harmonic functions. We show that $|Du|^{\frac{p-2+s}{2}}Du$ belongs to the Sobolev space $W^{1,2}_{loc}$, $s>-1-\frac{p-1}{n-1}$, for any $p$-harmonic function $u$. The proof is based on an elementary inequality.
We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of ${\mathbb R}^{n+1}$ with bounded energy. This generalizes results obtained by Da Lio on the disc.
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.
In the seminal paper [Arch. Ration. Mech. Anal. 176 (2005), 351--361], Savin proved the $C^1$-regularity of planar $\infty$-harmonic functions $u$. Here we give a new understanding of it from a capacity viewpoint and drop several high technique arguments therein. Our argument is essentially based on a topological lemma of Savin, a flat estimate by Evans and Smart, % \cite{es11a}, $W^{1,2}_{loc}$-regularity of $|Du|$ and Crandall's flow for infinity harmonic functions.
We present a proof of the local Hölder regularity of the horizontal derivatives of weak solutions to the $p$-Laplace equation in the Heisenberg group $\mathbb{H}^1$ for $p>4 $.
The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.
We prove that manifold constrained $p(x)$-harmonic maps are $C^{1,β}$-regular outside a set of zero $n$-dimensional Lebesgue's measure, for some $β\in (0,1)$. We also provide an estimate from above of the Hausdorff dimension of the singular set.
We prove the uniqueness of solutions to Dirichlet problem for p-harmonic maps with images in a small geodesic ball of the target manifold. As a consequence, we show that such maps have Hoelder continuous derivatives. This gives an extension of a result by S. Hildebrandt et al concerning harmonic maps.
Let $1<p\le4$ when $n=1$ and $1<p< 3+\frac{1}{n-1} $ when $n\ge2$. We obtain the second-order horizontal Sobolev $HW^{2,2}_{\rm loc} $-regularity of $p$-harmonic functions in the Heisenberg group $\mathbb H^n$. This improves the known range of $p$ obtained by Domokos and Manfredi in 2005.
We prove the regularity of weak 1/2-harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2-harmonic map equation in the form of a non-local linear Schrödinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.
In this paper, we study the fractional harmonic gradient flow on $S^1$ taking values in $S^{n-1} \subset \mathbb{R}^n$ for every $n \geq 2$, in particular addressing uniqueness and regularity of solutions in the so-called energy class with sufficiently small energy, adding to the existing body of knowledge which includes existence of solutions and certain bubbling phenomena.
For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model.
We prove a global version of the classical result that $p$-harmonic functions belong to $W^{2,2}_{loc}$ for $1<p<3+\frac{2}{n-2}$. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6].
We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,α}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the $\mathrm{L}^{p}$-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.
We consider the Navier-Stokes-Fourier system with the inhomogeneous boundary conditions for the velocity and the temperature. We show that solutions emanating from sufficiently regular data remain regular as long as the density $\varrho$, the absolute temperature $\vartheta$, and the modulus of the fluid velocity $|\textbf{u}|$ remain bounded.
We introduce n/p-harmonic maps as critical points of E(v) the Lp-Norm of the alpha-laplacian of v, where pointwise v maps Rn into a sphere, and alpha = n/p. This energy combines the non-local behaviour of the fractional harmonic maps introduced by Riviere and the first author with the degenerate arguments of the n-laplacian. In this setting, we will prove Holder continuity.
In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain $Ω\subset\R^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,...,m-1$, we will show $(u,v)$ is smooth.
In this paper, we continue to study the fractional harmonic gradient flow on $S^{n-1}$ taking values in a general closed manifold $N \subset \mathbb{R}^n$, addressing global existence and uniqueness of solutions of energy class with sufficiently small energy, adding to the existing body of knowledge pertaining to the half-harmonic gradient flow and expanding upon our previous work in [34]. We extend the techniques by Struwe in [30] and Rivière in [22] to the non-local framework analogous to [34] to derive uniqueness, employ commutator estimates as in [8] for regularity and follow [30] for a ge…
The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with $C^{1,1}$ boundary. We assume that at least one of the material parameters is $W^{1,3+δ}$ for some $δ>0$. Using regularity theory for second order elliptic partial differential equations, we derive $W^{1,p}$ estimates and Hölder estimates for electric and magnetic fields up to the boundary. We also derive interior estimates in bi-anisotropic media.
Given an arbitrary planar $\infty$-harmonic function $u$, for each $α>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^α$, which is sharp as $α\to0$. We also show that the distributional determinant of $u$ is a Radon measure enjoying some quantitative lower and upper bounds. As a by-product, for each $p>2$ we obtain some quantitative local $W^{1,p}$-estimates of $u$, and consequently, an $L^p$-Liouville property for $\infty$-harmonic functions in whole plane.
We prove an $ε$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets. Instead, we follow the spirit of the last-named author's recent work on free boundary harmonic maps …
In this paper we continue the investigation of the regularity of the so-called weak $\frac{n}{p}$-harmonic maps in the critical case. These are critical points of the following nonlocal energy \[ {\mathcal{L}}_s(u)=\int_{\mathbb{R}^n}| ( {-Δ})^{\frac{s}{2}} u(x)|^p dx\,, \] where $u\in \dot{H}^{s,p}(\mathbb{R}^n,\mathcal{N})$ and ${\mathcal{N}}\subset\mathbb{R}^N$ is a closed $k$ dimensional smooth manifold and $s=\frac{n}{p}$. We prove Hölder continuity for such critical points for $p \leq 2$. For $p > 2$ we obtain the same under an additional Lorentz-space assumption. The regularity theory i…
We study the regularity of weak solutions for two elliptic systems involving the $n$-Laplacian and a critical nonlinearity in the right hand side: $H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under the assumptions that the solutions belong to $W^{n/2,2}$ in an even dimension $n$, we prove their continuty. The tools used in the proof involve Hardy spaces and BMO, and the Rivière--Uhlenbeck decomposition (with estimates in Morrey spaces). A prominent role is played by the Coifman--Rochberg--Weiss commutator theorem.
This paper studies the regularity of Villani solutions of the space homogeneous Landau equation with Coulomb interaction in dimension 3. Specifically, we prove that any such solution belonging to the Lebesgue space L_{t}^{\infty}L_{v}^{q} with q>3 in an open cylinder (0,S)\times B, where B is an open ball of \mathbb{R}^{3}, must have Holder continuous second order derivatives in the velocity variables, and first order derivative in the time variable locally in any compact subset of that cylinder.
Let $(g^{αβ}(x))$ and $(h_{ij}(u))$ be uniformly elliptic symmetric matrices, and assume that $h_{ij}(u)$ and $p(x) \, (\, \geq 2)$ are sufficiently smooth. We prove partial regularity of minimizers for the functional [ {\mathcal F}(u) = \int_Ω(g^{αβ}(x) h_{ij}(u) D_αu^iD_βu^j)^{p(x)/2} dx, \] under the non-standard growth conditions of $p(x)$-type. If $g^{αβ}(x)$ are in the class $VMO$, we have partial Hölder regularity. Moreover, if $g^{αβ}$ are Hölder continuous, we can show partial $C^{1,α}$-regularity.
We prove continuity on domains up to the boundary for n/2-polyharmonic maps into manifolds. Technically, we show how to adapt Helein's direct approach to the fractional setting. This extends a remark by the author that this is possible in the setting of Riviere's famous regularity result for critical points of conformally invariant variational functionals. Moreover, pointwise behavior for the involved three-commutators is established. Continuity up to the boundary is then obtained via an adaption of Hildebrandt and Kaul's technique to the non-local setting.
In this article, we investigate the regularity for certain elliptic systems without a $L^2$-antisymmetric structure. As applications, we prove some $ε$-regularity theorems for weakly harmonic maps from the unit ball $B= B(m) \subset \mathbb{R}^m $ $(m\geq2)$ into certain pseudo-Riemannian manifolds: standard stationary Lorentzian manifolds, pseudospheres $\mathbb{S}^n_ν\subset \mathbb{R}^{n+1}_ν$ $(1\leqν\leq n)$ and pseudohyperbolic spaces $\mathbb{H}^n_ν\subset \mathbb{R}^{n+1}_{ν+1}$ $(0\leqν\leq n-1)$. Consequently, such maps are shown to be Hölder continuous (and as smooth as the regulari…
We prove Hoelder continuity for n/2-harmonic maps from subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Riviere to arbitrary dimensions. The proof relies on compensation effects which we quantify adapting an approach for Wente's inequality by L. Tartar, instead of Besov-space arguments which were used in the one-dimensional case. Moreover, fractional analogues of Hodge decomposition and higher order Poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied.
The Boltzmann equation is a nonlinear partial differential equation that plays a central role in statistical mechanics. From the mathematical point of view, the existence of global smooth solutions for arbitrary initial data is an outstanding open problem. In the present article, we review a program focused on the type of particle interactions known as non-cutoff. It is dedicated to the derivation of a priori estimates in $C^\infty$, depending only on physically meaningful conditions. We prove that the solution will stay uniformly smooth provided that its mass, energy and entropy densities rem…
We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior $ε$-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
Via Gauge theory, we give a new proof of partial regularity for harmonic maps in dimension m>2 into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of "minimal" C^2 regularity. The proof we present moreover extends to a large class of elliptic systems of quadratic growth.
We prove partial regularity for minimizers of quasiconvex functionals of the type $\int_Ωf(x,Du) dx$ with $p(x)$ growth with respect to the second variable. The proof is direct and uses a method of $A$-harmonic approximation.
This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and S. Nicaise, Tech. Rep. 10-09, IRMAR (2010)] for the case with homogeneous boundary condition. Unfortunately, their method can not be applied to the inhomogeneous case. The main novelty of this paper is that we follow the spirit of the $H^1$-estimate in [R. Hiptmair, A. Moiola and I. Perugia, Math. Models Methods Appl. Sci., 21(2011), pp. 2263-2287] and modi…
In this paper boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincaré inequality, but the results are new also in unweighted Euclidean spaces.
We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.
For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $ΣM$, and any compact Riemannian manifold $N$, we show an $ε$-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when $n = 2$. For $n \ge 3$, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in $R^n$. If, additions, $ψ\in W^{1,p}$ for some $p>2n/3$, th…
In this paper, we study the regularity of the solutions of Maxwell's equations in a bounded domain. We consider several different types of low regularity assumptions to the coefficients which are all less than Lipschitz. We first develop a new approach by giving $\mathcal{H}^1$ estimate when the coefficients are $\mathcal{L}^{\infty}$ bounded; and then we derive $\mathcal{W}^{1,p}$ estimates for every $p > 2$ when one of the leading coefficients is simply continuous; Finally, we extend the result to $\mathcal{C}^{1,α}$ almost everywhere for the solution of the homogeneous Maxwell's equations w…
We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hoelder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.
