Über die Regularität der schwachen Lösungen von Randwertaufgaben für quasilineare elliptische Differentialgleichungen höherer Ordnung
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data , where and , , or , or . Moreover, if , or if , or if and we show that the Cauchy problem is unconditionally wellposed in . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal...
We consider a class of stationary viscous Hamilton-Jacobi equations aswhere , is a bounded and uniformly elliptic matrix and is convex in and grows at most like , with and . Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate,i.e., for a certain (optimal) exponent . This completes the recent results in [15],...
We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation on Ω in the setting of the space H₀(Ω).
Consider the Navier-Stokes equation with the initial data . Let and be two weak solutions with the same initial value . If satisfies the usual energy inequality and if where is the multiplier space, then we have .
A generalized approach to several universality results is given by replacing holomorphic or harmonic functions by zero solutions of arbitrary linear partial differential operators. Instead of the approximation theorems of Runge and others, we use an approximation theorem of Hörmander.