For a given convex ring Omega = Omega(2)\(Omega) over bar (1) and an L-1 function f : Omega x R -> R+ we show, under suitable assumptions on f, that there exists a solution (in the weak sense) to Delta(p)u = f(x, u) in Omega u = 0 on partial derivative Omega(2) u = M on partial derivative Omega(1) with {x is an element of Omega : u(x) > s} boolean OR Omega(1) convex, for all s is an element of (0, M).

2. Surgery and the Spinorial tau-InvariantAmmann, Bernd

et al.

Dahl, Mattias

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Humbert, Emmanuel

Surgery and the Spinorial tau-Invariant2009In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 34, no 10, p. 1147-1179Article in journal (Refereed)

Abstract [en]

We associate to a compact spin manifold M a real-valued invariant (M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's sigma-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then (N) epsilon min{(M), n}, where n is a positive constant depending only on n=dim M. Various topological conclusions can be drawn, in particular that is a spin-bordism invariant below n. Also, below n the values of cannot accumulate from above when varied over all manifolds of dimension n.

We consider the parabolic obstacle type problem Hu = f chi(Omega) in Q(1)(-), u = vertical bar del u vertical bar = 0 on Q(1)(-)\Omega, where Omega is an unknown open subset of Q(1)(-). This problem has its origin in parabolic potential theory. When f is merely Holder continuous, the usual method based on the use of a monotonicity formula does not apply. Nevertheless, we can, under a combination of energetic and geometric assumptions, prove the optimal C-x(1,1) boolean AND C-t(0,1) regularity of the solution.

The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right.

We develop a method for automatically symmetrizing Petrowsky well-posed Cauchy problems for constant coefficient linear partial differential equations. The method is rooted in the Sturm sequence technique for establishing the location of the roots of a complex polynomial and can be automated using standard symbolic computation tools. In the special case of homogeneous strictly hyperbolic scalar equations, we show that the resulting estimates are strong enough to control all principal order derivatives and thus can be used in place of the Leray energies. We also illustrate the method by applying it to various problems of mixed type.

Let k and μk be the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure in Rd, d1. Filonov has proved in a simple way that the inequality μk+1<k holds for the Laplacian. We extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfill certain geometric conditions.

In this paper we study the homogenization of p-Laplacian with thin obstacle in a perforated domain. The obstacle is defined on the intersection between a hyperplane and a periodic perforation. We construct the family of correctors for this problem and show that the solutions for the epsilon-problem converge to a solution of a minimization problem of similar form but with an extra term involving the mean capacity of the obstacle. The novelty of our approach is based on the employment of quasi-uniform convergence. As an application we obtain Poincare's inequality for perforated domains.

9. Regularity of a free boundary at the infinity pointKarp, L.

et al.

Shahgholian, Henrik.

KTH, Superseded Departments, Mathematics.

Regularity of a free boundary at the infinity point2000In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 25, no 12-Nov, p. 2055-2086Article in journal (Refereed)

Abstract [en]

Suppose there is a nonnegative function u and an open set Ohm subset of R-n(n greater than or equal to 3), satisfying Deltau = chi (Ohm) in B-r(e), u = \delu\ = 0 on B-r(e)\Ohm, where B-r(e) = {x : \x\ > r}. Under a certain thickness condition on R-n\Ohm, we prove that the boundary of {x:x/\x\(2) is an element of Ohm} is a graph of a C-1 function in a neighborhood of the origin. As a by-product of the method of the proof, we also obtain the following result: Replace chi (Ohm) by f chi (Ohm), with a certain assumptions on f. Then for any solution u which is asymptotically nonnegative at infinity, there holds lim(r-->infinity) (\Br\)/(\Ohm boolean AND Br\) is an element of {1/2,1}.

10. The Hunter-Saxton System and the Geodesics on a Pseudosphere

We show that the two-component Hunter-Saxton system with negative coupling constant describes the geodesic flow on an infinite-dimensional pseudosphere. This approach yields explicit solution formulae for the Hunter-Saxton system. Using this geometric intuition, we conclude by constructing global weak solutions. The main novelty compared with similar previous studies is that the metric is indefinite.