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We prove that if $\\rho \\in L^q(\\mathbb{R}^N) $ for $q>2N$, the unique minimizer $u_\\rho$ is of class $W_{loc}^{2,2}(\\mathbb{R}^N)$. Moreover, if the norm of $\\rho$ is sufficiently small, the minimizer is a weak solution of the associated PDE\n  \\begin{equation}\\label{eq:BI-abs} \\tag{$\\mathcal{BI}$} -\\o"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1804.02355","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-06T16:56:04Z","cross_cats_sorted":[],"title_canon_sha256":"e591ce65a65ad0bd822c47ddc9c6a56921cacc415e0c6fb15bca62240cbdeec8","abstract_canon_sha256":"bd8feae518d71efe8b5b6cdbcfa15b95b93da2c12bdf2656140431ef688f03dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:09.315838Z","signature_b64":"GQ/rk8V2hrii/EAIuahnr7PRUqJf3UZu4ePuuWmM4ZGn6WZGlfvFQ5Ezp5AXGp5qy9Q2xOHLTMzS+W1pyUAmDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f44c6df09d9a1bd4a2f7d80f3bf68f5ff437e299b22cce70c0d8bb59a118d62","last_reissued_at":"2026-05-17T23:59:09.315264Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:09.315264Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the regularity of the minimizer of the electrostatic Born-Infeld energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Iacopetti, Denis Bonheure","submitted_at":"2018-04-06T16:56:04Z","abstract_excerpt":"We consider the electrostatic Born-Infeld energy \\begin{equation*} \\int_{\\mathbb{R}^N}\\left(1-{\\sqrt{1-|\\nabla u|^2}}\\right)\\, dx -\\int_{\\mathbb{R}^N}\\rho u\\, dx, \\end{equation*} where $\\rho \\in L^{m}(\\mathbb{R}^N)$ is an assigned charge density, $m \\in [1,2_*]$, $2_*:=\\frac{2N}{N+2}$, $N\\geq 3$. We prove that if $\\rho \\in L^q(\\mathbb{R}^N) $ for $q>2N$, the unique minimizer $u_\\rho$ is of class $W_{loc}^{2,2}(\\mathbb{R}^N)$. Moreover, if the norm of $\\rho$ is sufficiently small, the minimizer is a weak solution of the associated PDE\n  \\begin{equation}\\label{eq:BI-abs} \\tag{$\\mathcal{BI}$} -\\o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02355","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1804.02355","created_at":"2026-05-17T23:59:09.315371+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.02355v2","created_at":"2026-05-17T23:59:09.315371+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.02355","created_at":"2026-05-17T23:59:09.315371+00:00"},{"alias_kind":"pith_short_12","alias_value":"F5CMNXYJ3GQ3","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"F5CMNXYJ3GQ32SRP","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"F5CMNXYJ","created_at":"2026-05-18T12:32:22.470017+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X","json":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X.json","graph_json":"https://pith.science/api/pith-number/F5CMNXYJ3GQ32SRPPWAPHP3I6X/graph.json","events_json":"https://pith.science/api/pith-number/F5CMNXYJ3GQ32SRPPWAPHP3I6X/events.json","paper":"https://pith.science/paper/F5CMNXYJ"},"agent_actions":{"view_html":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X","download_json":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X.json","view_paper":"https://pith.science/paper/F5CMNXYJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.02355&json=true","fetch_graph":"https://pith.science/api/pith-number/F5CMNXYJ3GQ32SRPPWAPHP3I6X/graph.json","fetch_events":"https://pith.science/api/pith-number/F5CMNXYJ3GQ32SRPPWAPHP3I6X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X/action/storage_attestation","attest_author":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X/action/author_attestation","sign_citation":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X/action/citation_signature","submit_replication":"https://pith.science/pith/F5CMNXYJ3GQ32SRPPWAPHP3I6X/action/replication_record"}},"created_at":"2026-05-17T23:59:09.315371+00:00","updated_at":"2026-05-17T23:59:09.315371+00:00"}