We study a nonlinear system made up of an elliptic equation of blended singular/degenerate type and Poisson's equation with a lowly integrable source. We prove the existence of a weak solution in any space dimension and, chiefly, derive an improved $\mathcal{C}^{1,\text{log-Lip}}$-regularity estimate using tangential analysis methods. The system illustrates a sophisticated version of the proverbial thermistor problem and our results are new even in simpler modelling scenarios.
We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently close to $p_0=2$. We show that this phenomenon is driven by the classical Calderón-Zygmund constant. As a byproduct of our analysis we show that $C^{1,α}$ regularity improves up to $C^{1,1^-}$, when p is close enough to 2. This result we believe it is particularly interesting in higher dimensions $n>2,$ when optimal $C^{1,α}$ regularity is related to the optimal…
Quantitative homogenization and large-scale regularity for nondivergence-form equations under a critical ellipticity moment
SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite $d$th moment, where $d$ is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves l…
We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and yield new, optimal regularity criteria even in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get re…
In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determinant of the metric. We also give corresponding elliptic regularity results and characterizations of local conformal flatness in low regularity settings.
We establish maximal local regularity results of weak solutions or local minimizers of \[ \operatorname{div} A(x, Du)=0 \quad\text{and}\quad \min_u \int_ΩF(x,Du)\,dx, \] providing new ellipticity and continuity assumptions on $A$ or $F$ with general $(p,q)$-growth. Optimal regularity theory for the above non-autonomous problems is a long-standing issue; the classical approach by Giaquinta and Giusti involves assuming that the nonlinearity $F$ satisfies a structure condition. This means that the growth and ellipticity conditions depend on a given special function, such as $t^p$, $φ(t)$, $t^{p(x…
We prove sharp regularity results for a general class of functionals of the type $$ w \mapsto \int F(x, w, Dw) \, dx\;, $$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$ w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;, $$ with $0<ν\leq b(\cdot)\leq L $. This changes its ellipticity rate according to the geometry of the level set $\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new methods and proofs, that are suitable to build regularity theorems…
We study the regularity of the solutions to initial-boundary value problems for N-systems of the p-Laplacian type, in $n\geq 3$ space variables, with square-integrable external forces in the space-time cylinder. So, the ellipticity coefficient remains unbounded. The singular case $μ=0 $ is covered.
It is well known that a twice-differentiable real-valued function $W:\operatorname{GL}^+(n)\rightarrow\mathbb{R}$ on the group $\operatorname{GL}^+(n)$ of invertible $n\times n-$matrices with positive determinant is rank-one convex if and only if it is Legendre-Hadamard elliptic. Many energy functions arising from interesting applications in isotropic nonlinear elasticity, however, are not necessarily twice differentiable everywhere on $\operatorname{GL}^+(n)$, especially at points with non-simple singular values. Here, we show that if an isotropic function $W$ on $\operatorname{GL}^+(n)$ is t…
Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
Université Paris Cité, CNRS, Sorbonne Université（巴黎西岱大学、法国国家科学研究中心、索邦大学）；O. Mayer Mathematics Institute, Romanian Academy（罗马尼亚科学院 O. Mayer 数学研究所）；Univ Rennes, INSA Rennes, CNRS（雷恩大学、雷恩国立应用科学学院、法国国家科学研究中心）
The notion of $p$-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of $p$-ellipticity ensures higher regularity of solutions of such equations. In this work we extend the notion of $p$-ellipticity to second order elliptic systems. Recall that for systems, there is no single notion of ellipticity, rather a more complicated picture emerges with ellipticity conditions of varying strength such as the Legendre, Legendre-Hadamard and integral condi…
The goal of this article is to establish local Lipschitz continuity of weak solutions for a class of degenerated elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity condition is a natural generalization of the p-Laplace equation and more general quasilinear elliptic equations with polynomial or exponential type growth.
We establish a new theory of regularity for elliptic complex valued second order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the coefficients of the matrix $A$ satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as $L^p$-dissipativity. Precisely, the regularity result is a reverse Hölder condition for $L^p$ averages of solutions on interior balls, and serves as a replacement for the De Giorgi - Nash - Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studie…
In two dimensions every weak solution to a nonlinear elliptic system $\rm{div} a(x,u,Du)=0$ has Hölder continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent $p \geq 2$. We give an example showing that this result cannot be extended to the subquadratic case, i.e. that weak solutions are not necessarily continuous if $1< p <2$. Furthermore, we discuss related results for variational integrals.
We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is $C^{2,α}$ on the compliment of a closed set of Hausdorff dimension at most $ε$ less than the dimension. The equation is assumed to be $C^1$, and the constant $ε> 0$ depends only on the dimension and the ellipticity constants. The argument combines the $W^{2,ε}$ estimates of Lin with a result of Savin on the $C^{2,α}$ regularity of viscosity solutions which are close to quadratic polynomials.
We obtain weighted $L^2$ estimates for the elastic wave equation perturbed by singular potentials including the inverse-square potential. We then deduce the Strichartz estimates under the sole ellipticity condition for the Lamé operator $-Δ^\ast$. This improves upon the previous result in \cite{BFRVV} which relies on a stronger condition to guarantee the self-adjointness of $-Δ^\ast$. Furthermore, by establishing local energy estimates for the elastic wave equation we also prove that the solution has local regularity.
We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain $Ω$, \[ f_\zbar = {\cal A}(f_z) \hskip15pt a.e.\;\; z\in Ω, \] when ${\cal A}$ is linear at $\infty$. Namely $W^{1,1}_{loc}(Ω)$ solutions are $W^{2,2+ε}_{loc}(Ω)$. Here $ε>0$ depends explicitly on the ellipticity bounds of ${\cal A}$. The condition ``is linear at $\infty$'' is necessary - the result is false for the equation $f_\zbar = k|f_z|$, for any $0<k<1$, ($k=0$ is Weyl's lemma). We discuss the subsequent higher regularity implications for fully non-linear Beltr…
We prove Hölder regularity results for a class of nonlinear elliptic integro-differential operators with integration kernels whose ellipticity bounds are strongly directionally dependent. These results extend those in [9] and are also uniform as the order of operators approaches 2.
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;…
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a …
Given $L\geq 1$, we discuss the problem of determining the highest $α=α(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^α_{\rm loc}$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $α(L)\gtrsim {\rm exp}(-CL^β)$, for some $C, β\geq 1$ depending on the dimension $N\geq 3$. We show that in the non-local case, $α(L)\gtrsim L^{-1-δ}$ for all $δ>0$.
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the …
In this paper, we study the problem of shock reflection by a wedge, with the potential flow equation, which is a simplification of the Euler System. In the work of M. Feldman and G. Chen, the existence theory of shock reflection problems with the potential flow equation was established, when the wedge is symmetric w.r.t. the direction of the upstream flow. As a natural extension, we study non-symmetric cases, i.e. when the direction of the upstream flow forms a nonzero angle with the symmetry axis of the wedge. The main idea of investigating the existence of solutions to non-symmetric problems…
We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sections in these spaces under a generalized ellipticity condition on the underlying family. In particular, this implies a covariant version of Meyers-Serrin\rq{}s theorem on the whole $\mathsf{L}^p$-scale, for arbitrary Riemannian manifolds. Furthermore, we prove a new local ellipti…
We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate $p$-Poisson equation with a bounded source are locally of class $C^{p^\prime}=C^{1,\frac{1}{p-1}}$; this regularity is optimal.
We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only on ellipticity constants and dimension, such that if $u$ is a viscosity solution of $F(D^2u) = f(x) \in L^p$, then $u\in W^{1+\varepsilon,p}$, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that $F(D^2u) \in L^p \implies (-Δ)^θu \in L^p$, for a universal cons…
We prove regularity results for solutions of the equation \[div(< AXu,X u>^{(p-2)/2} AX u) = 0,\] $1<p<\infty$, where $X=(X_1,...,X_m)$ is a family of vector fields satisfying Hörmander's ellipticity condition, $A$ is an $m\times m$ symmetric matrix that satisfies degenerate ellipticity conditions. If the degeneracy is of the form \[λw(x)^{2/p}|ξ|^2\leq < A(x)ξ,ξ>\leq Λw(x)^{2/p}|ξ|^2,\] $w \in A_p$, then we show that solutions are locally Hölder continuous. If the degeneracy is of the form \[ k(x)^{-2/p'}|ξ|^2\leq < A(x)ξ,ξ>\leq k(x)^{2/p}|ξ|^2, \] $k\in A_{p'}\cap RH_τ$,where $τ$ depends on …
We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain $Ω\subset \mathbb{R}^2$ and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function ${\rm dist}(\cdot,\partial Ω)$. The results are extended to anisotropic problems without splitting structure under natural growth and elliptici…
We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic re…
For a parabolic equation associated to a uniformly elliptic operator, we obtain a $W^{3, \varepsilon}$ estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic $W^{2,\varepsilon}$ estimates with a comparison principle argument. As an application, we show, assuming the operator is $C^1$, that a viscosity solution is $C^{2,α}$ on the complement of a closed set of Hausdorff dimension $\varepsilon$ less than that of the ambient space, where the constant $\varepsilon>0$ depends only on the dime…
Parameter-ellipticity with respect to a closed subsector of the complex plane for pseudodifferential Douglis-Nirenberg systems is discussed and shown to imply the existence of a bounded H_\infty-calculus in suitable scales of Sobolev, Besov, and Hoelder spaces. We also admit non pseudodifferential perturbations. Applications concern systems with coefficients of mild Hoelder regularity and the generalized thermoelastic plate equations.
We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal Hessian power integrability $\varepsilon = \varepsilon(λ, Λ, n)$ in the celebrated $W^{2, \varepsilon}$-regularity estimate satisfies $$\frac{ \left (1+ \frac{2}{3}\left(1- \fracλΛ \right )\right )^{n-1}}{\ln n^4} \cdot \left( \fracλΛ \right) ^{n-1} \le \varepsilon \le \frac{nλ}{(n-1)Λ+λ}, $$ where $n\ge 3$ is the dimension and $0< λ< Λ$ are the ellipticity constants. In particular, $\left( \fracΛλ \right) ^{n-1} \varepsilon(…
An elliptic equation div(F(Du)) = f whose ellipticity strongly degenerates for small values of Du (say, F = 0 on B(0,1)) is considered. The aim is to prove regularity for F(Du). The paper proves a continuity result in dimension two and presents some applications.
In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the $p(x)$-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class $C^{1,κ}$ for a universal constant $0< κ< 1$. A key feature of our estimates is that they do not depend on the modulus of…
This paper addresses the gain of global fractional regularity in Nikolskii spaces for solutions of a class of quasilinear degenerate equations with $(p,q)$-growth. Indeed, we investigate the effects of the datum on the derivatives of order greater than one of the solutions of the $(p,q)$-Laplacian operator, under Dirichlet's boundary conditions. As it turns out, even in the absence of the so-called Lavrentiev phenomenon and without variations on the order of ellipticity of the equations, the fractional regularity of these solutions ramifies depending on the interplay between the growth paramet…
Regularity of solutions of the Navier-Stokes-αβ equations with wall-eddy boundary conditions
University of Florence（佛罗伦萨大学）；McGill University（麦吉尔大学）；National University of Mongolia（蒙古国立大学）；Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）
We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form $Lu=\text{div}(A \nabla u)-u_t=0$ in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A = \left[ a_{ij}(X,t)\right] $ is uniformly elliptic and bounded. We show that if the Regularity problem $(R)_p$ for the equation $Lu=0$ is solvable for some $1<p<\infty$ then the Dirichlet problem $(D^*)_{p'}$ for the adjoint equation $L^*v=0$ is also solvable, where $p'=p/(p-1)$. This result is an analogue of the result established in the elliptic case by Kenig and Pi…
We develop a maximal regularity approach in temporally weighted $L_p$-spaces for vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and conditions of Lopatinskii-Shapiro type are the basic structural assumptions. The weighted framework allows to reduce the initial regularity and to avoid compatibility conditions at the boundary, and it provides an inherent smoothing effect of the solutions. Our main tools are interpolation and trace theory for anisotropic Slobodetskii spaces with temporal wei…
In this paper we classify the nonnegative global minimizers of the functional \[ J_F(u)=\int_ΩF(|\nabla u|^2)+λ^2χ_{\{u>0\}}, \] where $F$ satisfies some structural conditions and $χ_D$ is the characteristic function of a set $D\subset \mathbb R^n$. We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\partial\{u>0\}$ can be seen as a rectifiable $n-1$ varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded and use Allard's monotonicity formula to prove the existen…
We prove that if u is a weak solution to a constant coefficient system (with strong ellipticity assumed along the horizontal direction) in a Carnot group (no restriction on the step), then u is actually smooth. We then use this result to develop blow-up analysis to prove a partial regularity result for weak solutions of certain non-linear systems.
We study existence and regularity of weak solutions for the following PDE $$ -\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;\mbox{in $B_1$}. $$ where $A(x,s) = A_+(x)χ_{\{s>0\}}+A_-(x)χ_{\{s\le 0\}}$ and $f(x,s) = f_+(x)χ_{\{s>0\}}+f_-(x)χ_{\{s\le 0\}}$. Under the ellipticity assumption that $\frac{1}μ\le A_{\pm} \le μ$, $A_{\pm}\in C(Ø)$ and $f_{\pm}\in L^N(Ø)$, we prove that under appropriate conditions the PDE above admits a weak solution in $W^{1,p}(B_1)$ which is also $C^{0,α}_{loc}$ for every $α\in (0,1)$ with precise estimates. Our methods relies on similar techniques as those deve…
