A COMPARISON PRINCIPLE FOR A SOBOLEV GRADIENT SEMI-FLOW

作者：Blass Timothy^*; De la Llave Rafael; Valdinoci Enrico

来源：Communications on Pure and Applied Analysis, 2011, 10(1): 69-91.

DOI：10.3934/cpaa.2011.10.69

• Comparison principle
• Sobolev gradient
• semigroups of linear operators
• fractional powers of elliptic operators

We consider gradient descent equations for energy functionals of the type S(u) = 1/2 < u(x), A(x)u(x)>(L2) + integral(Omega)V(x, u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration.
We consider the steepest descent equation for S where the gradient is an element of the Sobolev space H(beta), beta is an element of (0, 1), with a metric that depends on A and a positive number gamma > sup vertical bar V(22)vertical bar. We prove a weak comparison principle for such a gradient flow.
We extend our methods to the case where A is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.

• 出版日期2011-1