Doubly Nonlinear Equations of Porous Medium Type

作者：Boegelein Verena; Duzaar Frank; Marcellini Paolo; Scheven Christoph

来源：Archive for Rational Mechanics and Analysis, 2018, 229(2): 503-545.

DOI：10.1007/s00205-018-1221-9

In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given by
partial derivative(t)u(m) - div D-xi f (x, Du) = 0,
with , or more generally with an increasing and piecewise C (1) nonlinearity b and a function f depending on u
partial derivative(t)b(u) - div D-xi f (x, u, Du) = -D-u f(x, u, Du).
For the function f we merely assume convexity and coercivity, so that, for instance, with 1 < p < q and non-negative coefficients alpha, beta with , and are covered. Thus, for functions satisfying only a coercivity assumption from below but very general growth conditions from above, we prove the existence of variational solutions; mean while, if grows naturally when as a polynomial of order p (for instance in the case of the p-Laplacian operator), then we obtain the existence of solutions in the sense of distributions as well as the existence of weak solutions. Our technique is purely variational and we treat both the cases of bounded and unbounded domains. We introduce a nonlinear version of the minimizing movement approach that could also be useful for the numerics of doubly nonlinear equations.

• 出版日期2018-8