A Comparison Principle for a Sobolev Gradient Semi-Flow

We consider gradient descent equations for energy functionals of the type S(u) = 1/2 _{L^2} + \int_{\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 \in (0,1), with a metric that depends on A and a positive number \gamma > \sup |V_{22}|. 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. We 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

[1]  D. Prowe.  Berlin , 1855, Journal of public health, and sanitary review.

[2]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[3]  Milan Miklavčič,et al.  Applied Functional Analysis and Partial Differential Equations , 1998 .

[4]  Luis Silvestre,et al.  Regularity theory for fully nonlinear integro‐differential equations , 2007, 0709.4681.

[5]  H. P..  Annales de l'Institut Henri Poincaré , 1931, Nature.

[6]  K. Fernow.  New York , 1896, American Potato Journal.

[7]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[8]  J. Moser.  A stability theorem for minimal foliations on a torus , 1988, Ergodic Theory and Dynamical Systems.

[9]  S. Bochner.  Diffusion Equation and Stochastic Processes. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[10]  M. Shubin.  Pseudodifferential Operators and Spectral Theory , 1987 .

[11]  J. Moser.  A rapidly convergent iteration method and non-linear partial differential equations - I , 1966 .

[12]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[13]  J. Moser.  Minimal solutions of variational problems on a torus , 1986 .

[14]  C. Carracedo,et al.  The theory of fractional powers of operators , 2001 .

[15]  Tosio Kato,et al.  Note on Fractional Powers of Linear Operators , 1960 .

[16]  Damon Afkari,et al.  ? ? ? ? ? ? ? ? ? ? ? ? ? 30 ? ? ? ? ? ? ? ? ? ? ? ? ? ? , 2011 .

[17]  Michael E. Taylor,et al.  Partial Differential Equations , 1996 .

[18]  I. I. Vrabie.  C[0]-semigroups and applications , 2003 .

[19]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[20]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[21]  John William Neuberger,et al.  Sobolev gradients and differential equations , 1997 .

[22]  X. Cabré,et al.  Layer solutions in a half‐space for boundary reactions , 2005 .

[23]  R. Llave,et al.  A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations , 2009 .

[24]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[25]  G. Ragsdell.  Systems , 2002, Economics of Visual Art.

[26]  Michael E. Taylor,et al.  Partial Differential Equations III , 1996 .

[27]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[28]  Markus Haase,et al.  The Functional Calculus for Sectorial Operators , 2006 .

[29]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .