{"paper":{"title":"Sobolev regularity for a class of second order elliptic PDE's in infinite dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandra Lunardi, Giuseppe Da Prato","submitted_at":"2012-08-02T09:15:56Z","abstract_excerpt":"We consider an elliptic Kolmogorov equation $\\lambda u - Ku = f$ in a separable Hilbert space $H$. The Kolmogorov operator $K$ is associated to an infinite dimensional convex gradient system: $dX = (AX - DU(X))dt + dW (t)$, where $A $ is a self--adjoint operator in $H$ and $U$ is a convex lower semicontinuous function. Under mild assumptions we prove that for $\\lambda >0$ and $f\\in L^2(H,\\nu)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(H,\\nu)$, where $\\nu$ is the log-concave probability measure of the system. Moreover maximal estimates on the gradient of $u$ are proved. The ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0437","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}