Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact

Abstract

We investigate the regularity of the weak solution to elliptic transmission problems that involve several materials intersecting at a closed interior line of contact.We prove that local weak solutions possess second order generalized derivatives up to the contact line, mainly exploiting their higher regularity in the direction tangential to the line. Moreover we are thus able to characterize the higher regularity of the gradient and the Hölder exponent by means of explicit estimates known in the literature for two dimensional problems. They show that strong regularity properties, for instance the integrability of the gradient to a power larger than the space dimension d = 3, are to expect if the oscillations of the diffusion coefficient are moderate (that is for far larger a range than what a theory of small perturbations would allow), or if the number of involved materials does not exceed three.