Nonlinear flows and rigidity results on compact manifolds

This paper is devoted to rigidity results for some elliptic PDEs and to optimal constants in related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. The largest value of this parameter provides an estimate for the optimal constant in the corresponding interpolation inequality. Our approach relies on a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works on rigidity. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constants in the interpolation inequalities.     

Positive solutions of semilinear biharmonic equations with critical Sobolev exponents

In this paper we study the critical growth biharmonic problem with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the biharmonic problem.     

A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

We consider positive solutions of on B₁ (n?5) where μ and K>0 are smooth functions on B₁. If K is very sub-harmonic at each critical point of K in B2/3 and the maximum of u in is comparable to its maximum over , then all positive solutions are uniformly bounded on . As an application, a priori estimate for solutions of equations defined on Sn is derived.     

Uniform decay estimates for finite-energy solutions of semi-linear elliptic inequalities and geometric applications

We prove uniform decay estimates at infinity for solutions 0?u∈Lp of the semilinear elliptic inequality Δu+auσ+bu?0, a,b?0, σ?1, in the presence of a Sobolev inequality (with potential term). This gives a unified point of view in the investigation of different geometric questions. In particular, we present applications to the study of the topology at infinity of parallel mean curvature submanifolds, to the non-compact Yamabe problem, and to estimate the decay rate of the traceless Ricci tensor of conformally flat manifolds.     

Construction of singular limits for four-dimensional elliptic problems with exponentially dominated nonlinearity

The authors consider the existence of singular limit solution for a family of nonlinear elliptic problems with exponentially dominated nonlinearity and Navier boundary condition.     

On some degenerate quasilinear equations involving variable exponents

In the present paper, we are concerned with some degenerate quasilinear equations involving variable exponents. Using various (variational and nonvariational) techniques, we prove existence, nonexistence and multiplicity results.     

Double Aztec diamonds and the tacnode process

Discrete and continuous non-intersecting random processes have given rise to critical “infinite-dimensional diffusions”, like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary.     

Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

In this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established. The basic tools employed here are the Mountain Pass Theorem and the Concentration-Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation, whose respective associated functional is well defined in H¹(R^N) and satisfies the geometric hypotheses of the Mountain Pass Theorem.     

On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions

Research supported in part by NSF Grants DMS 8657483 and GER 9023335     

On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions

Summary We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domainsD ^d (d3), having compact boundary, with nonlinear Neumann boundary conditions on the boundary ofD. For this we use an implicit probabilistic representation, Schauder's fixed point theorem, and a recently proved Sobolev inequality forW ^1,2(D). Special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics.Research supported in part by NSF Grants DMS 8657483 and GER 9023335This article was processed by the authors using the style filepljourlm from Springer-Verlag.     

Radial entire solutions for supercritical biharmonic equations

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13. Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.     

Partial differential equations with differential constraints

A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints).     

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities.     

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).     

Systems of stochastic partial differential equations with reflection: Existence and uniqueness

In this paper, we establish the existence and uniqueness of solutions of systems of stochastic partial differential equations (SPDEs) with reflection in a convex domain. The lack of comparison theorems for systems of SPDEs makes things delicate.     

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we obtain a characterization of invariant measures of stochastic evolution equations and stochastic partial differential equations of pure jump type. As an application, it is shown that the equation has a unique invariant probability measure under some reasonable conditions.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.