Viscosity Solutions to Second Order Parabolic PDEs on Riemannian Manifolds

In this work we consider viscosity solutions to second order parabolic PDEs u _t +F(t,x,u,du,d ² u)=0 defined on complete Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold M has nonnegative sectional curvature, we get the finest results. If one additionally requires F to depend on d ² u in a uniformly continuous manner, the assumptions on curvature can be thrown away.     

Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators

We study uniformly elliptic fully nonlinear equations of the type F(D²u,Du,u,x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations.     

Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function evolve in such a way whenever u solves an equation u _t  + F(Du, D ² u) = 0, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off {Du = 0}, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D ² u. Finally we give some counterexamples showing that several well known properties of the evolutions in are no longer true when M has negative sectional curvature. D. Azagra was supported by grants MTM-2006-03531 and UCM-CAM-910626. M. Jimenez-Sevilla was supported by a fellowship of the Ministerio de Educacion y Ciencia, Spain. F. Macià was supported by program “Juan de la Cierva” and projects MAT2005-05730-C02-02 of MEC (Spain) and PR27/05-13939 UCM-BSCH (Spain).     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane

In this paper we study the monotonicity of positive (or non-negative) viscosity solutions to uniformly elliptic equations F(∇u,D²u)=f(u) in the half plane, where f is locally Lipschitz continuous (with f(0)?0) and zero Dirichlet boundary conditions are imposed. The result is obtained without assuming the u or |∇u| are bounded.     

Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

We consider the problem −Δu+a(x)u=f(x)|u|^2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|^2*−2u in Ω, u=0 on ∂Ω.     

Lane-Emden-Fowler equations with convection and singular potential

We are concerned with singular elliptic problems of the form −Δu±p(d(x))g(u)=λf(x,u)+μa|∇u| in Ω, where Ω is a smooth bounded domain in RN, d(x)=dist(x,∂Ω), λ>0, μ∈R, 0<a?2, and f is a nondecreasing function. We assume that p(d(x)) is a positive weight with possible singular behavior on the boundary of Ω and that the nonlinearity g is unbounded around the origin. Taking into account the competition between the anisotropic potential p(d(x)), the convection term a|∇u|, and the singular nonlinearity g, we establish various existence and nonexistence results.     

An inverse problem of identifying the coefficient in a nonlinear parabolic equation

This paper deals with the determination of a pair (p,u) in the nonlinear parabolic equation

u_t−u_xx+p(x)f(u)=0,     

Nonresonance between the first two eigenvalues for a Steklov problem

In this paper, we study the solvability of the Steklov problem Δ_pu=|u|^p−2u in Ω, on ∂Ω, under assumptions on the asymptotic behaviour of the quotients f(x,s)/|s|^p−2s and pF(x,s)/|s|^p which extends the classical results with Dirichlet boundary conditions that for a.e. x∈∂Ω, the limits at the infinity of these quotients lie between the first two eigenvalues.     

Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control

We study the Schrödinger equation i∂_tu+Δu+V₀u+V₁u=0 on R³×(0,T), where V₀(x,t)=|x-a(t)|^-1, with a∈W^2,1(0,T;R³), is a coulombian potential, singular at finite distance, and V₁ is an electric potential, possibly unbounded. The initial condition u₀∈H²(R³) is such that . The potential V₁ is also real valued and may depend on space and time variables. We prove that if V₁ is regular enough and at most quadratic at infinity, this problem is well-posed and the regularity of the initial data is conserved for the solution. We also give an application to the bilinear optimal control of the solution through the electric potential.     

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L¹-setting.     

Partial regularity in non-linear elasticity

We prove partial regularity of minimizers of some polyconvex functionals. In particular our results include models such as ∫_Ω a(x,u)(|Du|²+| det Du|²), where a is a bounded H?lder continuous function, such that a(x,u)≥c for some positive constant c. Received: 2 January 2001 / Revised version: 30 August 2001     

A further three critical points theorem

If X is a real Banach space, we denote by W_X the class of all functionals possessing the following property: if {u_n} is a sequence in X converging weakly to u∈X and lim inf_n→∞Φ(u_n)≤Φ(u), then {u_n} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C¹ functional, belonging to W_X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X^∗; a C¹ functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C¹ functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation

Φ^′(x)=λJ^′(x)+μΨ^′(x)     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

Comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule

A comparison principle for viscosity sub- and super-solutions of second order elliptic partial differential equations is derived using the “fuzzy sum rule” of non-smooth calculus. This method allows us to weaken the assumptions made on the function F when the equation F(x,u,Du,D²u)=0 is under consideration.     

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|^p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.     

Schauder estimates for a class of second order elliptic operators on a cube

We consider a class of second order elliptic operators on a d-dimensional cube S_d. We prove that if the coefficients are of class C^k+δ(S_d), with k=0,1 and δ∈(0,1), then the corresponding elliptic problem admits a unique solution u belonging to C^k+2+δ(S_d) and satisfying non-standard boundary conditions involving only second order derivatives.     

Large time behavior of the heat kernel

In this paper we study the large time behavior of the (minimal) heat kernel k_P^M(x,y,t) of a general time-independent parabolic operator Lu=u_t+P(x,∂_x)u which is defined on a noncompact manifold M. More precisely, we prove that

     

Heat kernel expansions in vector bundles

Let M be a complete connected smooth (compact) Riemannian manifold of dimension n. Let Π:V→M be a smooth vector bundle over M. Let be a second order differential operator on M, where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M. Let k_t(−,−) be the heat kernel of V relative to L. In this paper we will derive an exact and an asymptotic expansion for k_t(x,y₀) where y₀ is the center of normal coordinates defined on M, x is a point in the normal neighborhood centered at y₀. The leading coefficients of the expansion are then computed at x=y₀ in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M, of the vector bundle V, and of the vector bundle section ? and its derivatives.We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean-Singer).     

On improvement of summability properties of solutions of some degenerate nonlinear fourth-order equations

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. The right-hand side F?(x,?u) of the equations depends on the unknown function u. Under some conditions, including L^m -summability of F(·, 0) with m close to 1 and restrictions on the exponent σ of the growth of F?(x,?u) with respect to u, we establish existence of W-solutions of the Dirichlet problem for the given equations and describe the dependence of summability properties of these solutions on m and σ.