Bernhard Kawohl

Rearrangements and convexity of level sets in PDE

Rearrangements: un catalogue de rearrangements, proprietes communes des rearrangements, decroissance monotone et rearrangement quasiconcave, rearrangement symetrique decroissant, rearrangement decroissant monotone dans la direction y, rearrangement etoile, symetrisation de Steiner par rapport a {y=0}, symetrisation de Schwarz, symetrisation circulaire et spherique. Principes de maximum: methode du plan mobile, convexite des ensembles de niveau, concavite ou convexite des fonctions

Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas

Abstract The existence of global classical solutions to initial boundary value problems in the dynamics of a one-dimensional, viscous, heat-conducting gas is established. The nonlinear dissipative effects turn out to be sufficiently strong to prevent the development of singularities.

On Newton’s problem of minimal resistance

In 1685, Sir Isaac Newton studied the motion of bodies through an inviscid and incompressible medium. In his words (from his Principia Mathematica): If in a rare medium, consisting of equal particles freely disposed at equal distances from each other, a globe and a cylinder described on equal diameter move with equal velocities in the direction of the axis of the cylinder, (then) the resistance of the globe will be half as great as that of the cylinder.... I reckon that this proposition will be not without application in the building of ships.

Remarks on quenching

On considere le probleme aux limites parabolique non lineaire suivant: u t −Δ u =f(u), t>0, x∈Ω, u=0, t>0, n∈∂Ω, u(0, x)=u 0 (x)≥0, x∈Ω. On demontre un phenomene de trempe pour des non linearites non convexes dans un espace de dimension arbitraire

On rotationally symmetric mean curvature flow

The evolutionary motion of surfaces by their mean curvature has been studied by several authors from different points of view. A measure theoretic approach is studied by K. A. Brakke [B]; the classical parametric problem is studied by G. Huisken [Hl]. Non-parametric mean curvature evolution with boundary conditions is treated in [H2]. For a detailed discussion see [H3]. We restrict our attention to the case in which the initial surface has rotational symmetry. We shall show that under suitable initial conditions the solution degenerates in the sense that its curvature develops a singularity at exactly one point. This confirms numerical results of the first author [D] for general surfaces in R3. See also the work of M. A. Grayson [Gl, p. 286, G2]. The proofs rely on the parabolic maximum principle. The second author is gratefully indebted to S. Angencnt for bringing the problem to his attention and for pointing out the resemblance to so-called quenching problems. This research was financially via SFB 256 and SFB 123.

Gradient estimates for solutions of parabolic equations and systems

1. STATEMENT OF THE PROBLEM Let 52 c R”, n > 2, be a bounded domain with smooth boundary r= &2. In this note we study scalar quasilinear parabolic differential equations u, = div,Y(g( IVul’) VU) +f(u) = 0, (1) u,-A,u+f(u)=O (2) and systems u, DA,u + f(u) = 0 (3) on Q x (0, co), with initial conditions u( ., 0) = u0 resp. u( ., 0) = u0 and homogeneous Dirichlet boundary conditions u,i-X(O,CO)‘O for (1) and (2) resp. (4) u1r.x (0.00) = 0 for (3). (5)

Comparison Principle for Viscosity Solutions of Fully Nonlinear, Degenerate Elliptic Equations

We provide structural assumptions under which comparison principles hold for viscosity solutions of fully nonlinear degenerate elliptic equations. These assumptions appear to be more traceable and easier to verify than previously known ones.

Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions

This paper considers the dependence of the sum of the first m eigenvalues of three classical problems from linear elasticity on a physical parameter in the equation. The paper also considers eigenvalues

Remarks on eigenvalues and eigenfunctions of a special elliptic system

\gamma _i (a)

A geometric property of level sets of solutions to semilinear elliptic dirichiet problems

of a clamped plate under compression, depending on a lateral loading parameter