On the Interior Smoothness of Solutions to Second-Order Elliptic Equations

We study the interior smoothness properties of solutions to a linear second-order uniformly elliptic equation in selfadjoint form without lower-order terms and with measurable bounded coefficients. In terms of membership in a special function space we combine and supplement some properties of solutions such as membership in the Sobolev space W _{2, loc} ¹ and Holder continuity. We show that the membership of solutions in the introduced space which we establish in this article gives some new properties that do not follow from Holder continuity and the membership in W _2,loc ¹ .     

Convexity estimates for the solutions of a class of semi-linear elliptic equations

In this paper, we are concerned with convexity estimates for solutions of a class of semi-linear elliptic equations involving the Laplacian with power-type nonlinearities. We consider auxiliary curvature functions which attain their minimum values on the boundary and then establish lower bound convexity estimates for the solutions. Then we give two applications of these convexity estimates. We use the deformation method to prove a theorem concerning the strictly power concavity properties of the smooth solutions to these semi-linear elliptic equations. Finally, we give a sharp lower bound estimate of the Gaussian curvature for the solution surface of some specific equation by the curvatures of the domain's boundary.     

Schwarz symmetrization and comparison results for nonlinear elliptic equations and eigenvalue problems

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the existence and bound of solutions for some nonlinear equation. Moreover, for some nonlinear problems, we show that if the first p-eigenvalue of a domain is big, the supremum of a solution related to this domain is close to zero. For that we obtain L ^∞ estimates for solutions of nonlinear and eigenvalue problems in terms of other L ^p norms.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

Radial symmetry and monotonicity for an integral equation

In this paper we study radial symmetry and monotonicity of positive solutions of an integral equation arising from some higher-order semilinear elliptic equations in the whole space Rn. Instead of the usual method of moving planes, we use a new Hardy-Littlewood-Sobolev (HLS) type inequality for the Bessel potentials to establish the radial symmetry and monotonicity results.     

Boundary observability for the space-discretizations of the 1 — d wave equation

We consider the finite-difference and finite-element space discretization of the 1 — d wave equation with homogeneous Dirichlet boundary conditions in a bounded interval. We analyze the problem of estimating the total energy of solutions in terms of the energy concentrated on the boundary, uniformly as the net-spacing h → 0. We prove that there is no such a uniform bound due to spurious high frequencies. We prove however an uniform bound in suitable subspaces of solutions that eventually cover the whole energy space.     

Regularity of ground state solutions of dispersion managed nonlinear schrödinger equations

We consider the dispersion managed nonlinear Schrödinger equation (DMNLS) in the case of zero residual dispersion. Using dispersive properties of the equation and estimates in Bourgain spaces we show that the ground state solutions of DMNLS are smooth. The existence of smooth solutions in this case matches the well-known smoothness of the solutions in the case of nonzero residual dispersion. In the case x∈R² we prove that the corresponding minimization problem with zero residual dispersion has no solution.     

On Minimal Isotropic Tori In ℂ<Emphasis Type="Italic">P</Emphasis><Superscript>3</Superscript>

We show that one of the classes of minimal tori in CP³ is determined by the smooth periodic solutions to the sinh-Gordon equation. We also construct examples of such surfaces in terms of Jacobi elliptic functions.     

Existence and regularity of solution of mixed boundary value problem of a Busemann-equation with nonlinear absorb term

The Busemann-equation is a classical equation coming from fluid dynamics. The well-posed problem and regularity of solution of Busemann-equation with nonlinear term are interesting and important. The Busemann-equation is elliptic in y>0 and is degenerate at the line y=0 in R². With a special nonlinear absorb term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain. By means of elliptic regularization technique, a delicate prior estimate and compact argument, we show that the solution of mixed boundary value problem of Busemann-equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary on some conditions. The result is better than the result of classical boundary degenerate elliptic equation.     

Unbounded pseudodifferential calculus on Lie groupoids

We develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids (also called Lie groupoids) with compact basis. We analyze these operators as unbounded operators acting on Hilbert modules over C^∗(G), and we show in particular that elliptic operators are regular. We construct a scale of Sobolev modules which are the abstract analogues of the ordinary Sobolev spaces, and analyze their properties. Furthermore, we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense.     

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

For N-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in 1/r ^α with α ∈ (0, 2) we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three-body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.     

Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media

The Ginzburg-Landau-Allen-Cahn equation is a variational model for phase coexistence and for other physical problems. It contains a term given by a kinetic part of elliptic type plus a double-well potential. We assume that the functional depends on the space variables in a periodic way.We show that given a plane with rational normal, there are minimal solutions, satisfying the following properties. These solutions are asymptotic to the pure phases and are separated by an interface. The convergence to the pure phases is exponentially fast. The interface lies at a finite distance M from the chosen plane, where M is a universal constant. Furthermore, these solutions satisfy some monotonicity properties with respect to integer translations (namely, integer translations are always comparable to the function).We then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane-like properties.We also consider the case of possibly irrationally oriented planes. We show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions, one which is a minimizer and another one which is not. These solutions also have interfaces bounded by a universal constant, they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers.     

A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds

We extend a Liouville-type result of D. G. Aronson and H. F. Weinberger and E.N. Dancer and Y. Du concerning solutions to the equation Δ_pu=b(x)f(u) to the case of a class of singular elliptic operators on Riemannian manifolds, which include the ?-Laplacian and are the natural generalization to manifolds of the operators studied by J. Serrin and collaborators in Euclidean setting. In the process, we obtain an a priori lower bound for positive solutions of the equation in consideration, which complements an upper bound previously obtained by the authors in the same context.     

Nonlinear functionals of the Boltzmann equation and uniform stability estimates

We present nonlinear functionals measuring physical space variation and L¹-distance between two classical solutions for the Boltzmann equation with a cut-off inverse power potential. In the case that initial datum is a small, smooth perturbation of vacuum and decays fast enough in the phase space, we show that these functionals satisfy stability estimates which lead to BV-type estimates and a uniform L¹-stability.     

Elliptic equations,principal eigenvalue and dependence on the domain

We consider a general second order uniformly elliptic differential operator L and also the set θ of all open sets (not necessarily smooth) in the unit ball of ?ⁿ. We define a metric d in this space (up to an equivalence relation ～) that makes the space (θ/ ～,d) a complete metric space. We show that the principal eigenvalue and eigenfunction of L are continuous with the metric d. Similar results are obtained for the solutions of the equation Lv = ?.     

On the solutions of a functional equation of dhombres

Continuous solutions of the functional equation ?(x?(x)) = (?(x))² for x ∈ [0,∞) have been characterized by Dhombres. They form a simple, two-parametric family and the result can be easily extended to solutions on the whole real line. However, the class of all solutions is much larger. We show that there is a solution ? whose graph has, among others, the full outer two-dimensional Lebesgue measure. It turns out that partial restrictions, like boundedness, monotonicity and/or continuity on a subinterval do not imply the continuity of solutions. In particular, the class of monotonic solutions has interesting properties. We provide some techniques of constructing such solutions and show that, among others, there is a strictly increasing solution ? whose points of discontinuity form a dense set in R.     

Generalized Solutions to Semilinear Elliptic PDE with Applications to the Lichnerowicz Equation

In this article we investigate the existence of a solution to a semi-linear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semi-linear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori L ^∞-bounds and sub- and super-solutions to apply a fixed point argument. We then show that the net of solutions obtained through this process satisfies certain decay estimates by determining estimates for the sub- and super-solutions and utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. We motivate this Colombeau algebra framework by first solving an ill-posed critical exponent problem. To solve this ill-posed problem, we use a collection of smooth, “approximating” problems and then use the resulting sequence of solutions and a compactness argument to obtain a solution to the original problem. This approach is modeled after the more general Colombeau framework that we develop, and it conveys the potential that solutions in these abstract spaces have for obtaining classical solutions to ill-posed non-linear problems with irregular data.     

Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

     

On the structure of solutions to a class of quasilinear elliptic Neumann problems

We study the structure of positive solutions to the equation ?^mΔ_mu-u^m-1+f(u)=0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ?→0⁺ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ? for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1<m?2 holds and ? is sufficiently large, any positive solution must be a constant.