DIFFUSION COEFFICIENTS ESTIMATION FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

作者：Bonito Andrea^*; Cohen Albert; DeVore Ronald; Petrova Guergana; Welper Gerrit

来源：SIAM Journal on Mathematical Analysis, 2017, 49(2): 1570-1592.

DOI：10.1137/16M1094476

• parameter identification
• inverse problem
• elliptic partial differential equations
• stability

This paper considers the Dirichlet problem -div(a del u(a)) = f on D, u(a) = 0 on partial derivative D, for a Lipschitz domain D subset of R-d, where a is a scalar diffusion function. For a fixed f, we discuss under which conditions a is uniquely determined and when a can be stably recovered from the knowledge of u(a). A first result is that whenever a is an element of H-1(D), with 0 < lambda <= a <= A on D, and f is an element of L-infinity(D) is strictly positive, then ||a - b||(L2(D)) <= C||u(n) - u(b)||(1/6)(H01(D)). More generally, it is shown that the assumption a is an element of H-1(D) can be weakened to a is an element of H-s(D), for certain s < 1, at the expense of lowering the exponent 1/6 to a value that depends on s.

• 出版日期2017