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

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We consider second‐order elliptic transmission problems in 3D in which several subdomains intersect at a closed line of contact. We prove that weak solutions possess second‐order generalized derivatives up to the contact line. Moreover, we quantify the index of integrability and the Hölder exponent by means of explicit formula known in the literature for two‐dimensional problems. For instance, the integrability of the gradient to a power larger than the space dimension d=3 can be expected 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 there are at most three involved materials.