A. Aberqi, B. Aharrouch, J. Bennouna
We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain $Ω\subset \>I\!\!R^{N}\>(N \geq 2)\>$. We prove the existence of entropy solutions avoiding sign condition and coercivity on the lowers order terms.
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In this paper we consider an inverse coefficients problem for a quasilinear elliptic equation of divergence form $\nabla\cdot\vec{C}(x,\nabla u(x))=0$, in a bounded smooth domain $Ω$. We assume that $\overrightarrow{C}(x,\vec{p})=γ(x)\vec{p}+\vec{b}(x)|\vec{p}|^2+\mathcal{O}(|\vec{p}|^3)$, by expanding $\overrightarrow{C}(x,\vec{p})$ around $\vec{p}=0$. We give a reconstruction method for $γ$ and $\vec{b}$ from the Dirichlet to Neumann map defined on $\partialΩ$.
We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $Ω$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{ \begin{array}{ll} \mathcal A u+ Φ(u,\nabla u)=Ψ(u,\nabla u)+ \mathfrak{B} u \quad& \mbox{in } Ω,\\ u=0 & \mbox{on } \partialΩ. \end{array} \right. \end{equation} Here $ \mathcal A u=-\sum_{j=1}^N |\partial_j u|^{p_j-2}\partial_j u$ is the anisotropic $\overrightarrow{p}$-Laplace operator, while $\mathfrak B$ is an operator from $W_0^{1,\overrightarrow{p}}(Ω)$ into $W^{-1,\o…
Boundary differentiability is shown for solutions of nondivergence elliptic equations with unbounded drift
We proof a uniqueness and periodicity theorem for bounded solutions of uniformly elliptic equations in certain unbounded domains.
We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.
We prove global Holder estimates for solution of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions on the domain a' la Cabre'.
We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
This paper deals with a class of singularly perturbed nonlinear elliptic problems $(P_\e)$ with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as $\e\to 0$, and the domain is supposed to be unbounded and with unbounded boundary. Domains that enlarge at infinity, and whose boundary flattens or shrinks at infinity, are considered. It is proved that in such domains problem $(P_\e)$ has at least 2 solutions.
We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge-Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.
We prove heat kernel bounds for the operator (1 + |x|^α)Δin R^N, through Nash inequalities and weighted Hardy inequalities.
We prove global Sobolev regularity and pointwise upper bounds for the gradient of transition densities associated with second order differential operators in $\mathbb{R}^d$ with unbounded diffusion, drift and potential terms.
Using three different notions of generalized principal eigenvalue of linear second order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions satisfying Dirichlet boundary conditions. Relations between these principal eigenvalues, their simplicity and several other properties are further discussed.
Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of $L^\infty$-estimates on the second component of the system and is achieved by various comparison arguments.
We prove the one-dimensional symmetry of solutions to elliptic equations of the form $-\dive(e^{G(x)}a(|\nabla u|)\nabla u)=f(u) e^{G(x)}$, under suitable energy conditions. Our results hold without any restriction on the dimension of the ambient space.
This paper shows the unique solvability of elliptic problems associated with two-phase incompressible flows, which are governed by the two-phase Navier-Stokes equations, in unbounded domains such as the whole space separated by a compact interface and the whole space separated by a non-compact interface. As a by-product of the unique solvability of elliptic problems, we obtain the Helmholtz-Weyl decomposition for two-phase incompressible flows.
Maximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators $\mathcal{L}$ in $\mathbb{R}^N$ and we establish some criteria for an unbounded open set to be a Maximum Principle set for $\mathcal{L}$. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author).
We are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to $Du$. We prove several comparison principles among viscosity solutions which may be unbounded under some polynomial-type growth conditions. Our main result applies to PDEs with convex superlinear terms but we also obtain some results in nonconvex cases. Applications to monotone systems of PDEs are given.
We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.
In this paper, we use the variational method to find the normalized solutions of the quadratic coupled three wave Schrodinger equation with asymmetric coercive potential. We prove the existence of solutions for the system with 2-norm constraints.
Gradient estimates for quasilinear elliptic Neumann problems with unbounded first-order terms
本文在源项满足适当可积性条件下，针对一阶项关于梯度多项式增长的p-Laplacian型散度方程，建立了凸域上Neumann问题的全局先验梯度估计。
We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first-order, in the Lebesgue space L^p(R^d;R^m) with p in (1,\infty). Sufficient conditions to prove generation results of an analytic C_0-semigroup T(t), together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.
In this paper we prove that, under suitable assumptions on α > 0, the operator L = (1 + |x|α)Δadmits realizations generating contraction or analytic semigroups in Lp (RN). For some values of α, we also explicitly characterize the domain of L. Finally, some informations about the location and composition of the spectrum are given.
We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the Liouville-type theorem in the case when the coefficients may grow at the spacial infinity. We also study a corresponding parabolic problem in the $n$-dimensional whole space and discuss the energy identity for solutions in $L^q$. As a co…
Unbounded sets of solutions of non-cooperative elliptic systems on symmetric spaces
本文研究非合作椭圆系统在对称空间上的解集，证明非平凡解的连续族在平凡解集附近发散，利用不变强不定功能的度数理论和环面等价结构分析。
We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.
In this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in R^N, or in the exterior of a ball. We consider the case of fully coupled systems and nonlinearities which are either convex or have a convex derivative. The solutions are shown to be foliated Schwarz symmetric if a bound on their Morse index holds. As a consequence of the symmetry results we also obtain some nonexistence theorems.
We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers co…
We develop a new, unified approach to the following two classical questions on elliptic PDE: the strong maximum principle for equations with non-Lipschitz nonlinearities, and the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in…
We prove that, in a two-dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half-plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n-dimensional slabs in any dimension n.
This paper investigates quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, which continues the study of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682], where they investigated the limiting long time/large scale behavior of such a process under diffusive rescaling. We determine the effective equation and obtain the size estimates of the gradient of Green functions as well as the optimal (in general) convergence rates. The proof relies on transferring the non-divergence form into the divergence-form with the coe…
In this paper, we study the asymptotic behavior as $x_1\to+\infty$ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at $x_1=0$ is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space~$\mathbb{R}^N$. The second one is based on the theory of dynamical …
We develop an optimal regularity theory for $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly unbounded coefficients. Our approach treats both superlinear and sublinear gradient regimes in a unified way. We show $C^{0,α}$, $C^{0,\textrm{Log-Lip}}$, $C^{1,α}$, $ C^{1,\textrm{Log-Lip}}$ and $C^{2,α}$ regularity estimates, by displaying the growth allowed to the Hamiltonian in order to deal with an unbounded nonlinear gradient coefficient, whose integrabilit…
We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland [20] and Matano [44] states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain $Ω$ further satisfies $Ω$ $\cap$ {|x| $\le$ R} = O(R^2), when R $\rightarrow$ +$\infty$. If the domain is a straight cylinder, an additional natural assumption is needed. The…
We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying Ambrosetti-Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. [5], we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. [1].
Aleksandrov-Bakelman-Pucci maximum principle for $L^n$-viscosity solutions of uniformly elliptic equations with unbounded terms: The critical case
本文证明含无界漂移项的一致椭圆方程$L^n$-粘性解的ABP最大值原理，利用凸对偶与Trudinger-Moser不等式处理临界情形。
We extend the Caffarelli-Świech-Winter $C^{1,α}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. Instrumental in the proof are local pointwise bounds for the logarithmic gradient of the solution and its normal derivative, which we also establish.
Quasilinear elliptic problem in anisotrpic Orlicz-Sobolev space on unbounded domain
本文研究了全空间 $\mathbb{R}^n$ 上各向异性 Orlicz-Sobolev 空间中的拟线性椭圆问题，通过结合山路定理与 Lions 型引理证明了非平凡弱解的存在性，并推广了先前各项同性设定下的结果。
Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains
该研究探讨无界域几何对半线性椭圆方程解的影响，发现上镜区域的锥张角决定解的多寡，同时证明严格稳定解的唯一性，通过改进移动平面法和关联Delaunay曲面完成证明与构造。
We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $\mathcal Au+Φ(x,u,\nabla u)=\mathfrak{B}u+f$ in $Ω$, where $Ω$ is a bounded open subset of $\mathbb R^N$ and $f\in L^1(Ω)$ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $\mathcal A$, the prototype of which is $\mathcal A u=-\sum_{j=1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u)$ with $p_j>1$ for all $1\leq j\leq N$ and $\sum_{j=1}^N (1/p_j)>1$. As a novelty in this paper, our lower order terms involve a new class of operators $\mathfrak B…
A note on the weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
本文在局部 L^s 可积条件下证明 De Giorgi 类函数的弱 Harnack 不等式，将结果推广至非对数条件下的变指数与双相位广义 Orlicz 增长泛函。
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.
We prove that the realization $A_p$ in $L^p(\mathbb{R}^N),\,1<p<\infty$, of the elliptic operator $A=(1+|x|^α)Δ+b|x|^{α-1}\frac{x}{|x|}\cdot \nabla-c|x|^β$ with domain $D(A_p) =\{ u \in W^{2,p}(\mathbb{R}^N)\, |\, Au \in L^p(\mathbb{R}^N)\}$ generates a strongly continuous analytic semigroup $T(\cdot)$ provided that $α>2,\,β>α-2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [A.Canale, A. Rhandi, C. Tacelli, Ann. Sc. Norm. Super. Pisa CI. Sci. (5), 2016] and in [G.Metafune, C.Spina, C.Tacelli, Adv. Diff. Equat., 2014]. Moreover we show that $T(\cdot)$ is…
We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains $Ω\subset \mathbb R^n$ of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along the bounded directions of $Ω$, with possible degeneracy along the unbounded directions.
In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$, $\varepsilon>0$, can be determined from the full and partial Dirichlet-to-Neumann map. This extends the results from [M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam. (2021)] where this is shown for Hölder continuous potentials. Also we show that when the Dirichlet-to-Neumann map is restricted to one point on the boundary…
We prove rate of convergence results for singular perturbations of Hamilton-Jacobi equations in unbounded spaces where the fast operator is linear, uniformly elliptic and has an Ornstein-Uhlenbeck-type drift. The slow operator is a fully nonlinear elliptic operator while the source term is assumed only locally Hölder continuous in both fast and slow variables. We obtain several rates of convergence according on the regularity of the source term.
In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case.
We are concerned on the fourth-order elliptic equation \begin{equation}\tag{$P_λ$} \left\{ \begin{array}[c]{ll} Δ^2 u- Δu + V(x)u -λΔ[ρ(u^2)]ρ'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R}^N, & u\in W^{2,2}(\mathbb{R}^N), \end{array} \right. \end{equation} where $Δ^2 = Δ(Δ)$ is the biharmonic operator, $3\leq N\leq 6$, the radially symmetric potential $V$ may change sign and $\inf_{\mathbb{R}^N}V(x)=-\infty$ is allowed. If $f$ satisfies a type of nonquadracity and monotonicity conditions and $ρ$ is a suitable smooth function, we prove, via variational approach, the existence of a radially symme…
Keller-Osserman and Harnack type results for nonlinear elliptic PDE with unbounded ingredients
本文证明无界系数散度形式椭圆方程Keller-Osserman定理成立，解决强极大值原理开放问题，并推广Harnack不等式。
Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Unit Ball in $\mathbb R^3$ and Their Patterns
本文研究了三维单位球上半线性椭圆系统中非径向解的无界分支及其模式，通过等变度数和Burnside环技术，提出了一种检测和表征高维椭圆系统对称破坏的通用方法。
Boundedness of solutions to singular anisotropic elliptic equations
针对含奇异梯度依赖项的一般 Dirichlet 各向异性椭圆问题，在 $f\in L^r(Ω)$、$r>N/p$ 等条件下证明所有弱解一致有界。
Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction
该研究推广了已有线性情形的相关工作，刻画了柱形域长度趋于无穷时带混合边界条件的广义p-Laplace算子第一特征函数的渐近行为，还研究了线性情形高阶特征值、一般情形第二特征值的渐近行为。
Sign-changing solution for an overdetermined elliptic problem on unbounded domain
本文证明了无界域上超定椭圆问题变号解的存在性，通过建立高维核空间的Crandall-Rabinowitz型分岔定理克服了二维核空间困难，为Berenstein猜想提供了反例。
We provide conditions for existence of hyperbolic, unbounded periodic and elliptic solutions in terms of Weierstrass $\wp$ functions of both third and fifth-order KdV--BBM (Korteweg-de Vries--Benjamin, Bona \& Mahony) regularized long wave equation. An analysis for the initial value problem is developed together with a local and global well-posedness theory for the third-order KdV--BBM equation. Traveling wave reduction is used together with zero boundary conditions to yield solitons and periodic unbounded solutions, while for nonzero boundary conditions we find solutions in terms of Weierstra…
Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities
本文研究了涉及无界系数和奇非线性项的各向异性椭圆方程解的存在性和正则性。
In this paper we derive quantitative uniqueness estimates at infinity for solutions to an elliptic equation with unbounded drift in the plane. More precisely, let $u$ be a real solution to $Δu+W\cdot\nabla u=0$ in ${\mathbf R}^2$, where $W$ is real vector and $\|W\|_{L^p({\mathbf R}^2)}\le K$ for $2\le p<\infty$. Assume that $\|u\|_{L^{\infty}({\mathbf R}^2)}\le C_0$ and satisfies certain a priori assumption at $0$. Then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge \exp(-C_1R^{1-2/p}\log R)\quad\text{if}\quad 2<p<\infty \] and \[ \inf_…
Kernel estimates for parabolic systems of partial differential equations with unbounded coefficients
本文针对具有无界系数且扩散系数可能逐方程无界变化的非退化椭圆偏微分方程组，研究了其相关半群转移核的逐点上界估计。
On sublinear elliptic systems on bounded and thin unbounded domains
本文研究齐次Dirichlet边界条件下次线性增长弱耦合椭圆系统的正解存在性，推广了双方程次线性系统的现有结果，结论适用于有界与薄无界区域且可处理部分不连续非线性项。
A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms
本文研究了一类具有无界系数和奇异二次低阶项的非线性各向异性椭圆问题的存在性和正则性，通过分析模型问题得出了相关结论。
In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^α)Δ+b|x|^{α-1}\frac{x}{|x|}\cdot\nabla -|x|^β$ satisfies $$ k(t,x,y) \leq c_1e^{λ_0 t+ c_2t^{-γ}}\left(\frac{1+|y|^α}{1+|x|^α}\right)^{\frac{b}{2α}} \frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(β-α)}}{1+|y|^α} e^{-\frac{\sqrt{2}}{β-α+2}\left(|x|^{\frac{β-α+2}{2}}+ |y|^{\frac{β-α+2}{2}}\right)} $$ for $t>0,\,|x|,\,|y|\ge 1$, where $b\in\mathbb{R}$, $c_1,\,c_2$ are positive constants, $λ_0$ is the largest eigenvalue of the operator $A$, and $γ=\frac{β-α+2}{β+α-2}$, in the case where $N>2,\,α>2$ and $β>α-2$. The …
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