Sufficient conditions for the convexity of the level sets of ground-state solutions

Let Ω be a bounded convex domain in . We consider constrained minimization problems related to the Euler-Lagrange equation

Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Semilinear elliptic problems and concentration compactness on non-compact Riemannian manifolds

Existence of solution for semilinear problem with the Laplace-Beltrami operator on non-compact Riemannian manifolds with rich symmetries is proved by concentration compactness based on actions of the manifold's isometry group.     

The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, N ≥ 2, we show that positive solutions to the problem

Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifolds

We describe the asymptotic behaviour in Sobolev spaces of sequences of solutions of Paneitz-type equations [Eq. (E _α ) below] on a compact Riemannian manifold (M, g) which are invariant by a subgroup of the group of isometries of (M, g). We also prove pointwise estimates.     

Multiplicity theorems for superlinear elliptic problems

In this paper we study second order elliptic equations driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti–Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. Finally we indicate how in the semilinear case, Morse theory can be used to produce six nontrivial solutions. This paper was completed while the first author was visiting the University of Aveiro as an Invited Scientist. The hospitality and financial support of the host institution are gratefully acknowledged. The second and third authors acknowledge the partial financial support of the Portuguese Foundation for Science and Technology (FCT) under the research project POCI/MAT/55524/2004.     

Continuous families of Riemannian manifolds, Isospectral on functions but not on 1-forms

The purpose of this paper is to present the first continuous families of Riemannian manifolds that are isospectral on functions but not on 1-forms, and, simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. Examples of isospectral manifolds that are not isospectral on forms are sparse, as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral. The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method, a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method, and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson method. To the memory of Hubert Pesce, a valued friend and colleague.     

Some remarks on singular solutions of nonlinear elliptic equations. I

We study strong maximum principles for singular solutions of nonlinear elliptic and degenerate elliptic equations of second order. An application is given on symmetry of positive solutions in a punctured ball using the method of moving planes. Dedicated to Felix Browder on his 80th birthday     

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.     

Eigenvalues of $$(-\triangle + \frac{R}{2})$$ on manifolds with nonnegative curvature operator

In this paper, we show that the eigenvalues of are nondecreasing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat.     

Nonradial solutions for a conformally invariant fourth order equation in \mathbb {R}^4

We consider the following Liouville equation in

Perturbation from symmetry for indefinite semilinear elliptic equations

We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( ? s) =  ? g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\).     

Tubes and eigenvalues for negatively curved manifolds

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a² and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)²/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only. Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY Research Award Program.     

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

Generalized symmetric functions and invariants of matrices

We generalize the classical isomorphism between symmetric functions and invariants of a matrix. In particular, we show that the invariants over several matrices are given by the abelianization of the symmetric tensors over the free associative algebra. The main result is proved by finding a characteristic free presentation of the algebra of symmetric tensors over a free algebra. The author is supported by research grant Politecnico di Torino n.119, 2004.     

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains. Communicated by David Kinderlehrer     

Heat content and Brownian motion for some regions with a fractal boundary

LetD be an open, bounded set in euclidean space ^m (m=2, 3, ...) with boundary D. SupposeD has temperature 0 at timet=0, while D is kept at temperature 1 for allt>0. We use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofE _D (t), the amount of heat inD at timet, ast0⁺. For the triadic von Koch snowflakeK our results imply that

Highly oscillatory behavior of the activator in the Gierer and Meinhardt system

In this article we construct a new type of solutions for the Gierer and Meinhardt system

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006