Optimal boundary regularity for a singular Monge–Ampère equation

In this paper we study the optimal global regularity for a singular Monge–Ampère type equation which arises from a few geometric problems. We find that the global regularity does not depend on the smoothness of domain, but it does depend on the convexity of the domain. We introduce $(a,\eta )$ type to describe the convexity. As a result, we show that the more convex is the domain, the better is the regularity of the solution. In particular, the regularity is the best near angular points.     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

On certain degenerate and singular elliptic PDEs II: Divergence-form operators,Harnack inequalities,and applications

By means of the Monge–Ampère real-analysis and PDE techniques associated to certain convex functions, an approach towards Harnack inequalities is developed that simultaneously extends the one for uniformly elliptic operators from the De Giorgi–Nash–Moser theory and the one for the linearized Monge–Ampère operator from the Caffarelli–Gutiérrez theory. Applications include regularity properties for solutions to divergence-form elliptic equations with power-like singularities and $C^2$-estimates for solutions to the Monge–Ampère equation.     

A semilinear elliptic equation with a mild singularity at u = 0: Existence and homogenization

In this paper we consider singular semilinear elliptic equations whose prototype is the following

$\{\begin{array}{cc}?divA(x)Du=f(x)g(u)+l(x)\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$

where Ω is an open bounded set of ${\mathbb{R}}^N,N\ge 1$, $A\in L^{\infty }{(\mathrm{\Omega })}^{N\times N}$ is a coercive matrix, $g:[0,+\infty [\to [0,+\infty ]$ is continuous, and $0\le g(s)\le \frac{1}{s^{\gamma }}+1$ for every $s>0$, with $0<\gamma \le 1$ and $f,l\in L^r(\mathrm{\Omega })$, $r=\frac{2N}{N+2}$ if $N\ge 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x),l(x)\ge 0$ a.e. $x\in \mathrm{\Omega }$.We prove the existence of at least one nonnegative solution as well as a stability result; we also prove uniqueness if $g(s)$ is nonincreasing or “almost nonincreasing”.Finally, we study the homogenization of these equations posed in a sequence of domains ${\mathrm{\Omega }}^{\epsilon }$ obtained by removing many small holes from a fixed domain Ω.     

The Lp Robin problem for Laplace equations in Lipschitz and (semi-)convex domains

Let $n\ge 3$ and Ω be a bounded Lipschitz domain in ${\mathbb{R}}^n$. Assume that $p\in (2,\infty )$ and the function $b\in L^{\infty }(?\mathrm{\Omega })$ is non-negative, where ?Ω denotes the boundary of Ω. Denote by ν the outward unit normal to ?Ω. In this article, the authors give two necessary and sufficient conditions for the unique solvability of the Robin problem for the Laplace equation $\mathrm{\Delta }u=0$ in Ω with boundary data $?u/?\nu +bu=f\in L^p(?\mathrm{\Omega })$, respectively, in terms of a weak reverse Hölder inequality with exponent p or the unique solvability of the Robin problem with boundary data in some weighted $L^2(?\mathrm{\Omega })$ space. As applications, the authors obtain the unique solvability of the Robin problem for the Laplace equation in the bounded (semi-)convex domain Ω with boundary data in (weighted) $L^p(?\mathrm{\Omega })$ for any given $p\in (1,\infty )$.     

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

We study radial solutions of the semilinear elliptic equation

$\mathrm{\Delta }u+f(u)=0$

under rather general growth conditions on f. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole–Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $\mathrm{\Delta }u+u^p=0$ and $\mathrm{\Delta }u+e^u=0$.     

Convergence rates and interior estimates in homogenization of higher order elliptic systems

This paper is concerned with the quantitative homogenization of 2m-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\epsilon )$ convergence rate in $W^{m?1,p_0}$ with $p_0=\frac{2d}{d?1}$ in a bounded Lipschitz domain in ${\mathbb{R}}^d$ as well as the uniform large-scale interior $C^{m?1,1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m?1,1}$, $W^{m,p}$ and $C^{m?1,\alpha }$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.     

On integrally closed simple extensions of valuation rings

Let v be a Krull valuation of a field with valuation ring $R_v$. Let θ be a root of an irreducible trinomial $F(x)=x^n+a{x}^m+b$ belonging to $R_v[x]$. In this paper, we give necessary and sufficient conditions involving only $a,b,m,n$ for $R_v[\theta ]$ to be integrally closed. In the particular case when v is the p-adic valuation of the field $\mathbb{Q}$ of rational numbers, $F(x)\in \mathbb{Z}[x]$ and $K=\mathbb{Q}(\theta )$, then it is shown that these conditions lead to the characterization of primes which divide the index of the subgroup $\mathbb{Z}[\theta ]$ in $A_K$, where $A_K$ is the ring of algebraic integers of K. As an application, it is deduced that for any algebraic number field K and any quadratic field L not contained in K, we have $A_{KL}=A_K{A}_L$ if and only if the discriminants of K and L are coprime.     

Harnack inequality for quasilinear elliptic equations on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations

$\mathrm{div}\left(\frac{F^{\prime }(|\mathrm{?}u|)}{|\mathrm{?}u|}\mathrm{?}u\right)=h$

of p-Laplacian type on Riemannian manifolds, where an even function $F\in C^1\left(\mathbb{R}\right)\cap C^2(0,\infty )$ is supposed to be strictly convex on $(0,\infty )$. Under the assumption that either $F\in C^2(\mathbb{R})$ or its convex conjugate $F^{?}\in C^2(\mathbb{R})$ with some structural condition, we establish a (locally) uniform ABP type estimate and the Krylov–Safonov type Harnack inequality on Riemannian manifolds with the use of an intrinsic geometric quantity to the operator. Here, the $C^2$-regularities of F and $F^{?}$ account for degenerate and singular operators, respectively.     

Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field

${\epsilon }^{2s}{(?\mathrm{\Delta })}_{A/\epsilon }^{s}u+V(x)u=f(|u{|}^2)u\text{ in }{\mathbb{R}}^N,$

where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N\ge 3$, ${(?\mathrm{\Delta })}_A^s$ is the fractional magnetic Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ are continuous potentials and $f:{\mathbb{R}}^N\to \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for ε small.     

Logarithmic potentials on Pn

We study the projective logarithmic potential ${\mathbb{G}}_{\mu }$ of a probability measure μ on the complex projective space ${\mathbb{P}}^n$. We prove that the range of the operator $\mu ?{\mathbb{G}}_{\mu }$ is contained in the (local) domain of definition of the complex Monge–Ampère operator acting on the class of quasi-plurisubharmonic functions on ${\mathbb{P}}^n$ with respect to the Fubini–Study metric. Moreover, when the measure μ has no atom, we show that the complex Monge–Ampère measure of its logarithmic potential is an absolutely continuous measure with respect to the Fubini–Study volume form on ${\mathbb{P}}^n$.     

Regularity of random attractors for fractional stochastic reaction–diffusion equations on Rn

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in $H^s({\mathbb{R}}^n)$ with $s\in (0,1)$. We prove the existence and uniqueness of the tempered random attractor that is compact in $H^s({\mathbb{R}}^n)$ and attracts all tempered random subsets of $L^2({\mathbb{R}}^n)$ with respect to the norm of $H^s({\mathbb{R}}^n)$. The main difficulty is to show the pullback asymptotic compactness of solutions in $H^s({\mathbb{R}}^n)$ due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.     

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$. These groups contain as special cases the odd dimensional general linear groups ${\mathrm{GL}}_{2n+1}(R)$ where R is any ring, the odd dimensional orthogonal and symplectic groups ${\mathrm{O}}_{2n+1}(R)$ and ${\mathrm{Sp}}_{2n+1}(R)$ where R is any commutative ring and further the first author's even dimensional unitary groups ${\mathrm{U}}_{2n}(R,\mathrm{\Lambda })$ where $(R,\mathrm{\Lambda })$ is any form ring. We classify the E-normal subgroups of the groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ (i.e. the subgroups which are normalized by the elementary subgroup ${\mathrm{EU}}_{2n+1}(R,\mathrm{\Delta })$), under the condition that R is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ by conjugation on the set of all E-normal subgroups.     

Neeman's characterization of K(R-Proj) via Bousfield localization

Let R be an associative ring with unit and denote by $K(\mathrm{R}\text{-}\mathrm{Proj})$ the homotopy category of complexes of projective left R-modules. Neeman proved the theorem that $K(\mathrm{R}\text{-}\mathrm{Proj})$ is ${\mathrm{?}}_1$-compactly generated, with the category $K^{+}(\mathrm{R}\text{-}\mathrm{proj})$ of left bounded complexes of finitely generated projective R-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K(\mathrm{R}\text{-}\mathrm{Proj})$ vanishes in the Bousfield localization $K(\mathrm{R}\text{-}\mathrm{Flat})/〈K^{+}(\mathrm{R}\text{-}\mathrm{proj})〉$.     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

A generalised comparison principle for the Monge–Ampère equation and the pressure in 2D fluid flows

We extend the generalised comparison principle for the Monge–Ampère equation due to Rauch & Taylor (1977) [15] to nonconvex domains. From the generalised comparison principle, we deduce bounds (from above and below) on solutions to the Monge–Ampère equation with sign-changing right-hand side. As a consequence, if the right-hand side is nonpositive (and does not vanish almost everywhere), then the equation equipped with a constant boundary condition has no solutions. In particular, due to a connection between the two-dimensional Navier–Stokes equations and the Monge–Ampère equation, the pressure p in 2D Navier–Stokes equations on a bounded domain cannot satisfy $\mathrm{\Delta }p\le 0$ in Ω unless $\mathrm{\Delta }p\equiv 0$ (at any fixed time). As a result, at any time $t>0$ there exists $z\in \mathrm{\Omega }$ such that $\mathrm{\Delta }p(z,t)=0$.