Doubly Nonlinear Equations of Porous Medium Type
Doubly Nonlinear Equations of Porous Medium Type
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DOI：
10.1007/s00205-018-1221-9
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发表时间：
2018-08-01
影响因子：
2.5
通讯作者：
Scheven, Christoph
中科院分区：
文献类型：
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作者：
Boegelein, Verena;Duzaar, Frank;Scheven, Christoph
本文证明了一类双非线性方程解的存在性，其原型由偏导数(t)u(m) - div D-xi f(x, Du) = 0给出，具有或更一般地具有递增分段C(1)非线性b和依赖于上偏导数(t)b(u) - div D-xi f(x, u, Du) = - d -u f（x, u, Du）的函数f。对于函数f，我们仅仅假设它是凸性和矫顽力，因此，例如，当1 < p < q和非负系数α， β，和都被覆盖。因此，对于只满足下面的矫顽力假设而满足上面的非常一般的增长条件的函数，我们证明了变分解的存在性；同时，当它作为p阶多项式（如p-拉普拉斯算子）自然增长时，我们得到了在分布意义上解的存在性和弱解的存在性。我们的技术是纯变分的，我们处理有界和无界域的情况。我们介绍了一种非线性版本的最小化运动方法，它也可以用于双非线性方程的数值计算。
In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given bypartial 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 upartial 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.