Richard F. Bass
For $α\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+α}}$$ and for $α\in (0,1)$ we consider the same operator but where the $\nabla f$ term is omitted. We prove, under appropriate conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in $C^{α+β}$ if $f\in C^β$.
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We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$ {\rm d} X_t=σ(t, X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $( Z_t)_{t\geq 0}$ is a $d$-dimensional cylindrical $α$-stable process with $α\in (0, 2)$, $σ(t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is bounded $β$-Hölder continuous in $x$ uniformly in…
In this paper, we consider a product of a symmetric stable process in $\mathbb{R}^d$ and a one-dimensional Brownian motion in $\mathbb{R}^+$. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood-Paley functions which are obtained by the $α$-harmonic extension of an $L^p(\mathbb{R}^d)$ function.
We establish sharp regularity estimates for solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, being $L$ the generator of any stable and symmetric Lévy process. Such nonlocal operators $L$ depend on a finite measure on $S^{n-1}$, called the spectral measure. First, we study the interior regularity of solutions to $Lu=f$ in $B_1$. We prove that if $f$ is $C^α$ then $u$ belong to $C^{α+2s}$ whenever $α+2s$ is not an integer. In case $f\in L^\infty$, we show that the solution $u$ is $C^{2s}$ when $s\neq1/2$, and $C^{2s-ε}$ for all $ε>0$ when $s=1/2$. Then, we study the boundary regularity of solutions…
Let $2\le n\le 5$. We establish an apriori interior Hölder regularity of $C^2$-stable solutions to the semilinear equation $-Δu=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex in addition,we obtain an interior Hölder regularity, and hence the local boundedness, of $W^{1,2}(Ω)$-stable solutions by locally approximating them via $C^2(Ω)$-stable solutions. In particular, we do not require any lower bound on $f$.
We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-Δ)^su=e^u\quad\mathrm{in}\quad Ω\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$
The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue …
We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $α$-stable operator and the second one (possibly degenerate) corresponds to a class of \textit{lower order} Lévy measures. Such operators do not have a global scaling property. We establish Hölder regularity, Harnack inequality and boundary Harnack property of solutions of these operators.
Boundary regularity and Hopf lemma for nondegenerate stable operators
本文针对满足外部C^{1,dini}性质的有界开集，证明了非退化稳定积分微分算子方程解的尖锐边界Hölder正则性与Hopf型边界引理，并展示了估计在s→1-时的稳健性，从而恢复二阶方程经典结果。
We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equati…
Interior Regularity for Stable Solutions of Semilinear Elliptic Equations with Sign-Changing Nonlinearities
该研究针对具变号非线性项的半线性椭圆方程稳定解，在不同维数下建立了内部正则性估计，解决了相关公开问题并推广了已有结果，还证明了对应Liouville型定理。
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 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.
Boundary regularity for non-local operators with symmetric kernels and vanishing horizon
本文研究了具有奇异对称核且依赖于到域边界距离的非局部算子，证明了解的最优 Hölder 边界正则性及更高边界正则性。
We study a regularity property for the gain part of the relativistic Boltzmann collision operator. Our assumptions on the collisional scattering kernel cover the full range of generic hard and soft potentials with angular cut-off.
In a bounded domain, we consider a variable range nonlocal operator, which is maximally isotropic in the sense that its radius of interaction equals the distance to the boundary. We establish $C^{1,α}$ boundary regularity and existence results for the Dirichlet problem.
Regularity theory for mixed local-nonlocal problem involving general stable operators
本文针对含一般稳定Lévy型算子的混合局部-非局部椭圆方程，通过最大值原理和Liouville型结果，在适当假设下建立了解的内部及边界正则性。
Ultradifferentiable regularity properties of a class of hypoelliptic operators
研究一类由霍尔曼德引入的亚椭圆微分算子的超可微正则性，扩展了前人相关结果。
In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov-Backelman-Pucci estimate with 0<σ<2. And we show a Harnack inequality, Hölder regularity, and C^{1,α}-regularity of the solutions by obtaining decay estimates of their level sets.
Regularity of stable radial solutions to semilinear elliptic equations in MEMS problems
本文研究了MEMS问题中半线性椭圆方程稳定径向解的正则性，证明在特定条件下所有稳定径向解均为正则，解决了关于奇异非线性情况下稳定解正则性的开放问题。
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.
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 characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension $2$. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree $2$. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential…
Regularity of Structurally Stable Cusp Singularities for Two Families of Quasilinear Wave-type Equations
研究两类拟线性方程（Hunter-Saxton型和Camassa-Hom型）在参数λ∈(0,1)下尖点奇异性的正则性，给出I型和II型结构稳定奇异性的正则性，并证明λ→1时恢复C^{1/3}正则性。
We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-Δ)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$. Our main result establishes an $L^\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \leq n < 2(s+2+\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\in(0,1)$ in dimensions $2 \leq n\leq 6$. It applies to all nonlinearities $f\in C^2$. For such parameters $s$ and $n$, our result leads to the regularity of …
Regularity and Pohozaev identity for the Choquard equation involving the $p$-Laplacian operator
该研究针对含$p$-Laplacian算子的非线性Choquard方程，开展了弱解正则性分析并建立了Pohozaev型恒等式，丰富了相关方程的理论结果。
In this article we prove that solutions of singular fully nonlinear partial differential equations are $C^{1,β}$. We also prove the simplicity of the principal eigenvalues for the Dirichlet Problem associated to these operators using that regularity, a strict comparison principle and Sard's theorem.
In this paper, we mainly consider the Gevrey regularity and analyticity of the solution to a generalized two-component shallow water wave system with higher-order inertia operators, namely, $m=(1-\partial_x^2)^su$ with $s>1$. Firstly, we obtain the Gevrey regularity and analyticity for a short time. Secondly, we show the continuity of the data-to-solution map. Finally, we prove the global Gevrey regularity and analyticity in time.
Here we utilize operator--valued Lq-Lp Fourier multiplier theorems to establish lower bound estimates for large class of elliptic integro-differential equations in Rd. Moreover, we investigate separability properties of parabolic convolution operator equations that arise in heat conduction problems in materials with fading memory. Finally, we give some remarks on optimal regularity of elliptic differential equations and Cauchy problem for parabolic equations.
In this article we prove some Lipschitz estimates and existence result for a class of degenerate fully nonlinear elliptic equations which are a generalization of the pseudo p-Laplacian. The operators are degenerate elliptic at any point where even only one partial derivative of the solution is zero.
We obtain optimal continuity in Sobolev spaces for the Fourier integral operators associated to singular canonical relations, when one of the two projections is a Whitney fold. The regularity depends on the type, $k$, of the other projection from the canonical relation ($k=1$ for a Whitney fold). We prove that one loses $(4+\frac{2}{k})^{-1}$ of a derivative in the regularity properties. The proof is based on the $L^2$ estimates for oscillatory integral operators.
We prove that $C^{1,α}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.
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.
In this paper we deal with the problem of regularity for non hypo-elliptic partial differential equations with polynomial coefficients. An operator $A$ on on the space of tempered distributions $\mathcal{S}^\prime$ is regular if $u$ belongs to the Schwartz class $\mathcal S$ whenever $Au\in \mathcal S$. By using tensor products of topological vector spaces and a kind of Wigner transform we reduce the study of regularity of a non hypo-elliptic differential operator, like the twisted laplacian, to a hypo-elliptic one, like the harmonic oscillator.
This article is dedicated to the proof of the existence of classical solutions for a class of non-linear integral variational problems. Those problems are involved in nonlocal image and signal processing.
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.
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 $N > 2m$.
The main scope of this article is to define the concept of principal eigenvalue for fully non linear second order operators in bounded domains that are elliptic and homogenous. In particular we prove maximum and comparison principle, Holder and Lipschitz regularity. This leads to the existence of a first eigenvalue and eigenfunction and to the existence of solutions of Dirichlet problems within this class of operators.
We show a multi-anisotropic Gevrey regularity of solutions of hypoelliptic equations. This result is a precision of a classical result of Hörmander
We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-Hölder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of th…
In this paper we prove the higher Sobolev regularity of minimisers for convex integral functionals evaluated on linear differential operators of order one. This intends to generalise the already existing theory for the cases of full and symmetric gradients to the entire class of $\mathbb{C}$-elliptic operators therein including the trace-free symmetric gradient for dimension $n \geq 3$.
This note seeks to prove the existence of a canonical solution operator to the $\bar\partial$-equation that preserves Hölder regularity on product domains. It is a well known fact that such solution operators do not in general gain Hölder regularity, and as such, our solution operator is optimal in this regard.
A regularity theory for evolution equations with time-measurable pseudo-differential operators in weighted mixed-norm Sobolev-Lipschitz spaces
该研究针对带时间可测伪微分算子的发展方程，在加权混合范数Sobolev-Lipschitz空间框架下分析解的存在性、唯一性、正则性，并探究迹嵌入与解的连续性，丰富了相关方程的正则性理论。
We introduce a calculus for parameter-dependent singular Green operators on the half-space $\mathbb{R}^n_+$ that combines both elements of Grubb's calculus for boundary value problems of finite regularity and techniques of Schulze's calculus for pseudodifferential operators on manifolds with edges.
We establish the maximal regularity for nonautonomous Ornstein-Uhlenbeck operators in $L^p$-spaces with respect to a family of invariant measures, where $p\in (1,+\infty)$. This result follows from the maximal $L^p$-regularity for a class of elliptic operators with unbounded, time-dependent drift coefficients and potentials acting on $L^p(\R^N)$ with Lebesgue measure.
We establish higher-order weighted Sobolev and Holder regularity for solutions to variational equations defined by the elliptic Heston operator, a linear second-order degenerate-elliptic operator arising in mathematical finance. Furthermore, given $C^\infty$-smooth data, we prove $C^\infty$-regularity of solutions up to the portion of the boundary where the operator is degenerate. In mathematical finance, solutions to obstacle problems for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.
We examine the system given by \hfill -Δu = λ(v+1)^p \qquad Ω \hfill -Δv = γ(u+1)^θ\qquad Ω, \hfill u = v =0 \qquad \quad \partial Ω, where $ λ,γ$ are positive parameters and where $ 1 < p \le θ$ and where $ Ω$ is a smooth bounded domain in $ R^N$. We show the extremal solutions associated with the above system are bounded provided [\frac{N}{2} < 1 + \frac{2(θ+1)}{pθ-1} (\sqrt{\frac{p θ(p+1)}{θ+1}} + \sqrt{\frac{p θ(p+1)}{θ+1} - \sqrt{\frac{p θ(p+1)}{θ+1}}})]
The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $L^{p'}$, subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $L^p$. Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with $t…
We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.
We show maximal $L^p$-regularity for non-autonomous Cauchy problems provided the trace spaces are stable in some parameterized sense and the time dependence is of bounded variation. In particular, on $L^2$, we obtain for all $p \in (1,2]$ maximal $L^p$-regularity for non-autonomous elliptic operators with measurable coefficients.
Nonlocal degenerate Isaacs operators: Hölder regularity
本文研究非局部退化伊萨克算子的霍尔德正则性，证明在s接近1时解的霍尔德连续性，并应用得到利奥维勒定理。
In this note, we prove $\mathcal{C}^{1,γ}$ regularity for solutions of some fully nonlinear degenerate elliptic equations with "superlinear" and "subquadratic " Hamiltonian terms. As an application, we complete the results of \cite{BDL1} concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.
We prove an epsilon-regularity theorem for critical and super-critical systems with a non-local antisymmetric operator on the right-hand side. These systems contain as special cases, Euler-Lagrange equations of conformally invariant variational functionals as Rivière treated them, and also Euler-Lagrange equations of fractional harmonic maps introduced by Da Lio-Rivière. In particular, the arguments presented here give new and uniform proofs of the regularity results by Rivière, Rivière-Struwe, Da-Lio-Rivière, and also the integrability results by Sharp-Topping and Sharp, not discriminating be…
In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff-Backelman-Pucci estimate corresponding to the full class $\cS^{\fL_0}$ of uniformly elliptic nonlinear equations with $1<σ<2$ (subcritical case) and to their subclass $\cS^{\fL_0}_η$ with $0<σ\leq 1$. We show that $\cS^{\fL_0}_η$ still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and $C^{1,α}$-regularity of the solutions by obtaining decay estimates …
We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated $C^{1,α}$-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are base…
In this paper we obtain an energy estimate for a complete strictly hyperbolic operator with second order coefficients satisfying a log-Zygmund-continuity condition with respect to $t$, uniformly with respect to $x$, and a log-Lipschitz-continuity condition with respect to $x$, uniformly with respect to $t$.
Hölder regularity of stable solutions to semilinear elliptic equations up to $\mathbf{\mathbb{R}^9}$: full quantitative proofs
本文为半线性椭圆方程稳定解在最优维数 n≤9 内的 Hölder 正则性提供完整自足证明，并用较新的定量证明替代原紧性论证，以量化正则性指数并简化边界正则性处理。
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,α}$, then they are $C^\infty$. This completes the study of regular points, initiated in [5]. In order to achieve this, we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in $C^{k,α}$ domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for $k=\infty$) and by the second author and Serra (for $k=1$).
We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times Ω$, with $I \subset \mathbb{R}$ and $Ω\subset\mathbb{R}^n$. The operators $L$ we consider are infinitessimal generators of stable Lévy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that $u$ is $C^{2s+α}$ in $x$ and $C^{1+\fracα{2s}}$ in $t$, whenever $f$ is $C^α$ in $x$ and $C^{\fracα{2s}}$ in $t$. In the case $f\in L^\infty$, we prove that $u$ is $C^{2s-ε}$ in $x$ and $C^{1…
Weak solutions to parabolic integro-differential operators of order $α\in (α_0, 2)$ are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to $α\nearrow 2$. In this sense, the presentation is an extension of Moser's result in 1971.
This article proves the regularity for the Boltzmann equation without angular cutoff with hard potential. By sharpening the coercivity and upper bound estimate on the collision operator, analyzing the Poisson bracket between the transport operator and some weighted pseudo-differential operator, we prove the regularizing effect in space and velocity variables when the initial data has mild regularity.
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