{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:OURRV5PWADLICZJSNSFPN4DF34","short_pith_number":"pith:OURRV5PW","schema_version":"1.0","canonical_sha256":"75231af5f600d68165326c8af6f065df2daf0f9991a2bb5ca6c96cf41a015457","source":{"kind":"arxiv","id":"1707.01001","version":1},"attestation_state":"computed","paper":{"title":"Parabolic Regularity and Dirichlet boundary value problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luke Dyer, Martin Dindo\\v{s}","submitted_at":"2017-07-03T14:45:38Z","abstract_excerpt":"We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form $Lu=\\text{div}(A \\nabla u)-u_t=0$ in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A = \\left[ a_{ij}(X,t)\\right] $ is uniformly elliptic and bounded.\n  We show that if the Regularity problem $(R)_p$ for the equation $Lu=0$ is solvable for some $1<p<\\infty$ then the Dirichlet problem $(D^*)_{p'}$ for the adjoint equation $L^*v=0$ is also solvable, where $p'=p/(p-1)$.\n  This result is an analogue of the result established in the elliptic case by Kenig 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