Viscosity solutions for doubly nonlinear evolution equations

Abstract

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop.

Keywords:

• Viscosity solution
• partial differential inclusions
• doubly nonlinear equations
• vanishing viscosity limit
• minimizing movement
• hysteresis

2010 Mathematics Subject Classifications:

• 35D40
• 35G31
• 35K55
• 34E15
• 34A60
• 34C55

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

LC acknowledges support from the Luxembourg National Research Fund (FNR) (13502370).