{"id":"914af80e-e754-4879-8ec0-80c2b73035d2","arxiv_id":"2411.16377","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For bounded C^2 convex domains and p>1, the first weighted p-Laplace eigenfunction is log-concave and its first eigenvalue satisfies λ(Ω_t) ≤ (1-t)λ(Ω_0)+tλ(Ω_1).","lead":"This paper studies a diffusion-and-drift problem on rounded, bulging shapes and proves that its gentlest vibration pattern is a single smooth bump whose logarithm is convex. It also shows that the corresponding vibration frequency satisfies a Brunn-Minkowski-type rule when two shapes are blended together.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Brunn-Minkowski proof invokes a convex inf-convolution analogue of Lemmas 5.1 and 5.2 that is neither stated nor proved; if this p-Laplacian comparison fails, Theorem 1.2 does not follow.","rationale":"The reader's weakest assumption is exactly the unproved transfer of Lemmas 5.1 and 5.2 from sup-convolution of concave functions to inf-convolution of convex functions. I independently checked the algebra around that transfer, and the likely repair is available: for positive definite A,B, the harmonic mean H is Loewner-dominated by the arithmetic mean, which gives the displayed matrix inequality for all p>1 by the argument tr(A_bar-H) >= n^T(A_bar-H)n >= (2-p)n^T(A_bar-H)n when p<2, and the obvious monotonicity when p>=2. So the concern is a real gap in exposition and proof, but not a demonstrated falsehood. I did not find a fatal counterexample to Theorem 1.1 or 1.2; the strategy follows the established [11] method and the p=2 case is supported by the cited recent work. The manuscript also has numerous transcription errors and a confusing v_epsilon sign convention in Section 4, so the paper requires revision before it is acceptable, matching the reader's CONDITIONAL assessment. My read therefore does not move the verdict.","tokens_in":62,"tokens_out":31369,"duration_ms":352122,"concrete_test":"Independently derive the convex analogue of Lemma 5.2: for strictly convex C^2 functions w0,w1, set tilde w(z)=inf{(1-t)w0(x)+t w1(y): z=(1-t)x+ty}, let A=D^2w0(x), B=D^2w1(y), and H=[(1-t)A^{-1}+tB^{-1}]^{-1}. Verify for every p>1 and every unit vector n that tr(H)+(p-2)n^T H n <= (1-t)[tr(A)+(p-2)n^T A n] + t[tr(B)+(p-2)n^T B n]. If this holds, complete the missing proof that Delta_p tilde w(z) <= (1-t)Delta_p w0(x)+tDelta_p w1(y). If it fails for any p in (1,2), exhibit explicit A,B,n and use that example to display a failure of (5.12).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 1.1 and 1.2 both pass through limiting and comparison arguments, but the single weakest link is in Theorem 1.2. Lemmas 5.1 and 5.2 are quoted from [11] for strictly concave functions and sup-convolution. In the proof of Theorem 1.2, however, w_i=-ln u_i are convex, and the paper forms the inf-convolution (5.5). The needed convex analogue, obtained by conjugating the w_i as the remark (5.2) gestures at, is a Hessian formula H = [(1-t)A^{-1}+tB^{-1}]^{-1} and the comparison Delta_p tilde w(z) <= (1-t)Delta_p w0(x)+tDelta_p w1(y). This is asserted, not derived. The comparison is exactly what produces the term lambda_t in (5.12); if the implied matrix inequality F_p(H) <= (1-t)F_p(A)+tF_p(B), with F_p(M)=tr(M)+(p-2)n^T M n, fails for some p in (1,2), Theorem 1.2 collapses. There are also sign/notation inconsistencies in Section 4: v_epsilon is sometimes -ln u_epsilon and sometimes ln u_epsilon, and the equation fed into Korevaar's principle must be matched to the correct sign convention before Theorem 1.1 can be checked line by line.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the Gaussian-weighted p-Laplace eigenvalue problem (1.1) on bounded convex domains with C^2 boundary, with operator Δ_{p,γ}u = div(|∇u|^{p-2}∇u) - (x,∇u)|∇u|^{p-2}. After proving existence, positivity, uniqueness and C^{1,α} regularity (Theorem 3.1), the paper claims that the first positive eigenfunction is log-concave (Theorem 1.1) and that the first eigenvalue satisfies the Brunn-Minkowski-type inequality λ_{p,γ}(Ω_t) ≤ (1-t)λ_{p,γ}(Ω_0) + tλ_{p,γ}(Ω_1) for C^2 convex bodies (Theorem 1.2). The proof of Theorem 1.1 uses a regularized variational problem, interior C^{1,β}/C^{2,β} estimates, and Korevaar's concavity maximum principle; the proof of Theorem 1.2 uses log-concavity and the inf-convolution of the convex functions w_i = -ln u_i, following the strategy of Colesanti-Cuoghi-Salani.","tokens_in":19377,"tokens_out":8484,"duration_ms":75429,"significance":"If the proofs can be completed, Theorem 1.1 is a natural extension of known log-concavity results for p-Laplace eigenfunctions (Sakaguchi, Barles) to the Gaussian-weighted operator, and Theorem 1.2 would be a new convexity result for the weighted first eigenvalue in the class of convex bodies. The paper is honest about the main difficulty, namely that the equation is only weakly regular, and it attempts a regularization route rather than assuming higher regularity. It also correctly identifies that the non-homogeneity of the eigenvalue prevents the usual normalization in the Brunn-Minkowski inequality. The main value is in the two theorem statements and the methodological template; the manuscript does not contain reproducible code or machine-checked proofs, but that is not expected for this type of result.","major_comments":[{"comment":"The proof of Theorem 1.2 invokes, without proof, an inf-convolution analogue of Lemmas 5.1 and 5.2. The lemmas as quoted from [11] are stated for strictly concave functions and for sup-convolution, whereas the functions w_i = -ln u_i are convex and the construction (5.3)–(5.5) is an inf-convolution. The remark (5.2) records the convex-duality identity (-u)^* = (1-t)(-u_0)^* + t(-u_1)^*, but the paper does not derive the transformed Hessian formula D^2 w(z) = [(1-t)A^{-1} + tB^{-1}]^{-1} nor the p-Laplacian comparison Δ_p w(z) ≤ (1-t)Δ_p w_0(x) + tΔ_p w_1(y). Inequality (5.12) is precisely where this comparison is used, so Theorem 1.2 is not established as written. If the comparison fails for some p in (1,2), the argument collapses; at minimum a proof or a precise citation of the convex/inf-convolution version is required.","section":"Section 5 (Lemmas 5.1–5.2 and proof of Theorem 1.2)"},{"comment":"The sign and definition of υ_ε are inconsistent across Section 4. Proposition 4.8 defines υ_ε = -ln u_ε and studies c_ε = υ_ε((1-t)x+ty) - (1-t)υ_ε(x) - tυ_ε(y) in order to prove convexity of -ln u_ε, while Proposition 4.9 states that υ_ε = ln u_ε is concave in Ω_ν and Proposition 4.10 and Theorem 1.1 conclude that υ = ln u is concave. Since concavity of ln u is equivalent to convexity of -ln u, the two conventions can be reconciled, but as written the equation in Proposition 4.8, with the coefficient displayed as (ε+|∇υ_ε|)^{(p-2)/2}, is not shown to be the equation satisfied by -ln u_ε under either sign convention, and the statement of Proposition 4.9 uses the opposite sign from Proposition 4.8. The limiting argument (4.17)–(4.20) and the application of Korevaar's principle cannot be checked line by line until this is fixed.","section":"Section 4 (Propositions 4.8–4.10 and Theorem 1.1)"},{"comment":"The comparison principle is not proved as written. After passing to the unweighted form, the proof states ∫ e^{-|x|^2/2}|∇u_1|^{p-2}∇u_1·∇ϕ ≤ ∫ e^{-|x|^2/2}|∇u_1|^{p-2}∇u_1·∇ϕ, with the same function u_1 on both sides; the subsequent test-function computation uses u_1 and u_2, indicating a typo, but the displayed inequality is tautological. Since Proposition 3.3 is used in the proof of Hopf's lemma (Proposition 3.5) and in the uniqueness argument in Theorem 3.1, the regularity and uniqueness part of Theorem 1.1 depends on a corrected proof.","section":"Proposition 3.3"},{"comment":"In the proof of Theorem 1.1, the passage from strongly convex domains Ω_k to Ω relies on the assertion that 'the C^β(Ω)-estimate of the solution to (1.1) is independent of small smooth perturbation of the boundary ∂Ω'. This is not proved and is not automatic, since the C^β constants for solutions on varying domains may depend on the boundary geometry. A reference or explicit argument is needed to justify the uniform bound ‖u_k‖_{C^β(Ω_k)} ≤ c used before applying Arzelà-Ascoli.","section":"Proof of Theorem 1.1 (approximation by strongly convex domains)"}],"minor_comments":[{"comment":"The displayed definition of Ω_t reads Ω_t = (1-t)Ω_0 + Ω_1 and is missing the factor t on Ω_1; it should be Ω_t = (1-t)Ω_0 + tΩ_1, as used throughout the proof.","section":"Theorem 1.2 statement"},{"comment":"In (5.9) and in the definition of F_ε, the second second-order term in each bracket should involve the unit vector n_i rather than n_{i,ε}; as printed, both occurrences are written with n_{i,ε}, so the ε-dependence does not cancel in the intended way.","section":"Equation (5.9) and definition of F_ε"},{"comment":"The text says 'by (5.7) and (5.7)' where the second reference should be (5.8), and in the surrounding sentences '∇w_{i,ε}' is sometimes written as '∇w_i,ε'; these notational slips should be corrected.","section":"Proof of Theorem 1.2, after (5.14)"},{"comment":"Equation (3.2) appears to have a typo on the right-hand side: the test function should be ψ, not ∇ψ, in the integral λ∫|u|^{p-2}u e^{-|x|^2/2}ψ dx; otherwise the weak formulation does not match (2.4).","section":"Equation (3.2)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal. The main risk is not novelty but completeness: the inf-convolution comparison in Section 5 and the sign/definition inconsistencies in Section 4 are load-bearing, and Proposition 3.3 needs a corrected proof. I recommend major revision, with the expectation that the author supply a self-contained proof or precise reference for the convex analogue of Lemma 5.2 and clean up the sign convention before the claims can be verified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this is a seriously intended extension of Sakaguchi's log-concavity theorem to the Gaussian-weighted p-Laplace operator for every p>1, together with a Brunn-Minkowski inequality for the first eigenvalue. The log-concavity part is the stronger half and is probably right; the BM part is plausible but as written does not close.\n\nWhat is genuinely new: the unweighted p-Laplace case is Sakaguchi, the weighted Ornstein-Uhlenbeck case is only p=2 (Colesanti–Francini–Livshyts–Salani), and the all-p weighted extension is not in the literature. The paper also correctly identifies that the lack of homogeneity prevents the usual weak BM statement, and it gives a full existence–uniqueness–regularity section, including Hopf and comparison lemmas adapted to the Gaussian weight. That is useful groundwork.\n\nWhere the soft spots are, in proportion:\n\n1. The BM proof has a load-bearing gap. Lemmas 5.1 and 5.2 quoted from [11] are stated for strictly concave functions and sup-convolution. In Theorem 1.2 the paper applies them to the convex functions w_i = -ln u_i and uses inf-convolution. The remark (5.2) gestures at the convex-duality route, but the actual statement and proof of the inf-convolution analogue—the Hessian formula and the pointwise comparison for Delta_p tilde w—are not given. Without that comparison, the inequality (5.12) has no foundation. I think the analogue is likely true, but it needs to be derived, not asserted.\n\n2. The manuscript is full of sign and transcription errors in exactly the places a referee would need to check. Theorem 1.2 defines Omega_t = (1-t)Omega_0 + Omega_1, dropping the factor t. In Proposition 3.3 the comparison inequality has the same integrand on both sides (u1 instead of u2 on the right). In Section 4, upsilon_epsilon is sometimes -ln u_epsilon and sometimes ln u_epsilon; Proposition 4.9 concludes \"upsilon_epsilon = ln u_epsilon is concave\" from a proof that works with the opposite sign. These are individually fixable, but they make line-by-line verification of Theorem 1.1 unnecessarily hard.\n\n3. Minor: the convergence argument for F_epsilon and the boundary term in (5.14) are standard but sketched; a referee will want more details on the passage to the limit in the singular set C_t.\n\nWho this is for: people working on convexity of solutions to degenerate elliptic equations and on Brunn–Minkowski inequalities for spectral quantities. They will get a clear statement of a natural open problem's likely answer, but they should not take the BM proof as finished.\n\nMy recommendation: send it to a serious referee. The result is important enough, and Theorem 1.1 is close to being solid. The referee should be asked to request a self-contained proof of the inf-convolution lemma and a full correction of the sign conventions before the paper is accepted.","headline":"Plausible extension of log-concavity to the Gaussian p-Laplace for all p>1, but the Brunn-Minkowski proof leans on an unproved convex inf-convolution lemma and the manuscript needs a careful sign cleanup.","tokens_in":19972,"tokens_out":3463,"would_cite":false,"duration_ms":33971,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P30","35J92","52A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on a bounded C^2 convex domain, the first positive eigenfunction of the weighted p-Laplace operator is log-concave, and that the first eigenvalue obeys a Brunn-Minkowski-type inequality under Minkowski combinations…","keywords":["log-concavity","weighted p-Laplace operator","Brunn-Minkowski inequality","first eigenvalue","Gaussian measure","convex bodies","quasilinear elliptic equations","concavity maximum principle"],"falsifier":"Compute $\\lambda_{p,\\gamma}$ numerically for two concentric balls in $\\mathbb R^2$ (or a ball and an ellipse) for several $p>1$, and check whether $\\lambda_{p,\\gamma}(\\Omega_t)\\le(1-t)\\lambda_{p,\\gamma}(\\Omega_0)+t\\lambda_{p,\\gamma}(\\Omega_1)$; a violation for any $C^2$ convex bodies refutes Theorem 1.2. For Theorem 1.1, solve (1.1) on a $C^2$ convex domain with a flat boundary segment and inspect the Hessian of $w=-\\ln u$ at points where $\\nabla u\\neq0$; one negative eigenvalue of $D^2w$ refutes log-concavity.","tokens_in":18876,"feed_emoji":"📐","tokens_out":12937,"duration_ms":107261,"temperature":0.7,"pith_summary":"This paper proves two convexity statements for the Gaussian-weighted $p$-Laplace eigenvalue problem on bounded convex domains with smooth boundary. The first is that the positive first eigenfunction $u$ of $\\Delta_{p,\\gamma}u:=\\operatorname{div}(|\\nabla u|^{p-2}\\nabla u)-\\langle x,\\nabla u\\rangle |\\nabla u|^{p-2}$ is log-concave: $w=-\\ln u$ is convex in $\\Omega$ for every $p>1$. The second is that the first eigenvalue $\\lambda_{p,\\gamma}$ is convex under Minkowski addition of $C^2$ convex bodies, meaning $\\lambda_{p,\\gamma}(\\Omega_t)\\le(1-t)\\lambda_{p,\\gamma}(\\Omega_0)+t\\lambda_{p,\\gamma}(\\Omega_1)$ for $\\Omega_t=(1-t)\\Omega_0+t\\Omega_1$. Together these extend the classical log-concavity of Laplacian eigenfunctions and Brunn-Minkowski-type inequalities to a degenerate quasilinear weighted setting.","feed_headline":"Eigenfunctions log-concave; eigenvalue convex under Minkowski sums","feed_subtitle":"The first eigenfunction is log-concave and the first eigenvalue obeys a Brunn-Minkowski-type inequality.","key_machinery":"The load-bearing objects are the weighted $p$-Laplace operator $\\Delta_{p,\\gamma}u=\\operatorname{div}(|\\nabla u|^{p-2}\\nabla u)-\\langle x,\\nabla u\\rangle |\\nabla u|^{p-2}$, the log-transform $w=-\\ln u$, and the inf-convolution of the convex functions $w_i$ on the Minkowski sum $\\Omega_t=(1-t)\\Omega_0+t\\Omega_1$. On the eigenfunction side, the argument runs through a regularized variational problem whose solutions are smooth in the interior, to which a concavity maximum principle applies; uniform $C^{1,\\beta}$ and $C^{2,\\beta}$ estimates let the regularization converge to the weak solution and carry convexity of the regularized logarithm to $w$. On the eigenvalue side, the Brunn-Minkowski inequality is transported via inf-convolution: the paper defines $\\tilde w(x)=\\inf\\{(1-t)w_0(x_0)+tw_1(x_1):x=(1-t)x_0+tx_1\\}$, shows $\\tilde w$ is $C^1$, obtains the comparison $\\Delta_p\\tilde w\\le(1-t)\\Delta_p w_0+t\\Delta_p w_1$ in the appropriate sense, and converts this into a bound on the Rayleigh quotient of the trial function $e^{-\\tilde w}$. The transfer relies on convex duality results from [22] and on comparison lemmas from [11] originally stated for strictly concave sup-convolutions, applied here to the convex $w_i$ after an implicit duality step.","core_discovery":"The central claim of the paper is that log-concavity and Brunn-Minkowski convexity survive the passage from the linear Laplacian to the weighted $p$-Laplace operator. On a bounded convex domain with $C^2$ boundary, the first positive eigenfunction of (1.1) has the property that $w=-\\ln u$ is convex, so all positive superlevel sets of $u$ are convex. On the class of $C^2$ convex bodies, the first eigenvalue satisfies $\\lambda_{p,\\gamma}((1-t)\\Omega_0+t\\Omega_1)\\le(1-t)\\lambda_{p,\\gamma}(\\Omega_0)+t\\lambda_{p,\\gamma}(\\Omega_1)$, a Brunn-Minkowski-type inequality for the eigenvalue map. The proof route is to establish existence, uniqueness, Hopf boundary behavior, and global $C^{1,\\alpha}$ regularity for the weak solution; to prove log-concavity through a regularized problem and a concavity maximum principle; and then to derive the eigenvalue inequality by inf-convolution of the convex functions $w_i=-\\ln u_i$ together with a comparison argument for the $p$-Laplacian of the inf-convolution.","pith_inferences":["A radial numerical test is immediately available: for concentric balls the Minkowski combination is a ball, so the inequality $\\lambda_{p,\\gamma}(B_R)\\le(1-t)\\lambda_{p,\\gamma}(B_r)+t\\lambda_{p,\\gamma}(B_s)$ with $R=(1-t)r+ts$ can be checked from the one-dimensional radial eigenfunction problem; a violation would refute Theorem 1.2.","The same inf-convolution strategy should work for other convex weights $e^{-V}$ replacing the Gaussian density, since the drift term enters only through the first-order term; testing $V=|x|^4/4$ would separate Gaussian-specific properties from the general convex-geometric mechanism.","The approximating sequence $\\Omega_k\\uparrow\\Omega$ in Theorem 1.1 is where strong convexity is dropped, so a $C^2$ convex domain with a flat boundary segment is the natural place to look for counterexamples if log-concavity does not survive without strong convexity.","Writing out the duality step between sup-convolutions and inf-convolutions explicitly would turn Section 5 into a general comparison theorem for convex solutions of (5.1), making the Brunn-Minkowski inequality independent of the regularity machinery."],"forward_implications":["Every positive superlevel set $\\{u>c\\}$ of the first eigenfunction is convex whenever $\\Omega$ is convex, so Brunn-Minkowski-type measure bounds apply to these level sets.","The eigenvalue map $\\Omega\\mapsto\\lambda_{p,\\gamma}(\\Omega)$ is convex along Minkowski linear combinations, giving the nonlinear analogue of the Brunn-Minkowski inequalities already known for torsional rigidity and $p$-capacity.","For $p=2$, Theorems 1.1 and 1.2 recover the log-concavity and Brunn-Minkowski results for the Gaussian Laplacian (Ornstein-Uhlenbeck) eigenvalue problem recently obtained by other methods.","The comparison theory developed here (weak comparison principle, Hopf boundary lemma, uniqueness up to scale) supplies tools for further variational problems for the weighted $p$-Laplacian on Gaussian spaces."],"supporting_citations":[{"why":"supplies the two comparison lemmas for sup-convolutions of strictly concave functions that Section 5 transfers to the inf-convolution setting.","marker":"[11]"},{"why":"supplies the concavity maximum principle applied to the regularized eigenfunctions to obtain convexity of $\\ln u_\\varepsilon$.","marker":"[17]"},{"why":"provides the convex-analysis facts (inf-convolution regularity, subdifferential calculus, conjugates) used to build $\\tilde w$ and identify its critical set.","marker":"[22]"},{"why":"provides the interior $C^{1,\\alpha}$ regularity estimates for quasilinear elliptic equations used for the eigenfunction and its regularizations.","marker":"[24]"},{"why":"supplies the boundary regularity method via Schwarz reflection that gives global $C^{1,\\alpha}$ estimates.","marker":"[25]"},{"why":"supplies the local $C^{1,\\alpha}$ regularity theory for degenerate elliptic equations used in the regularity section.","marker":"[3]"},{"why":"supplies Schauder estimates and the Hopf and strong maximum principles used for boundary behavior and uniqueness proofs.","marker":"[15]"},{"why":"supplies the variational existence and bootstrap regularity framework for weak solutions of the eigenvalue problem.","marker":"[18]"}],"fun_headline_variants":["Weighted p-Laplace: eigenfunctions log-concave, eigenvalues convex","Log-concavity and Brunn-Minkowski for weighted p-Laplace","Eigenfunction log-concavity persists in weighted p-Laplace","Weighted p-Laplace: log-concave eigenfunctions, BM convex eigenvalues"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The second theorem rests on assuming without proof that the comparison lemmas for combining strictly concave functions still hold for the inf-convolutions of the convex functions $w_i=-\\ln u_i$; if that transfer fails, the Brunn-Minkowski inequality does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Weighted p-Laplace: eigenfunctions log-concave, eigenvalues convex","Log-concavity and Brunn-Minkowski for weighted p-Laplace","Eigenfunction log-concavity persists in weighted p-Laplace","Weighted p-Laplace: log-concave eigenfunctions, BM convex eigenvalues"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000687,"raw_usage":{"total_tokens":3073,"prompt_tokens":860,"completion_tokens":2213,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":476,"completion_tokens_details":{"reasoning_tokens":2129}},"tokens_in":476,"tokens_out":2213,"duration_ms":148639,"temperature":1.0,"reasoning_tokens":2129,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:13:03.627700+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\lambda_{p,\\gamma}$ numerically for two concentric balls in $\\mathbb R^2$ (or a ball and an ellipse) for several $p>1$, and check whether $\\lambda_{p,\\gamma}(\\Omega_t)\\le(1-t)\\lambda_{p,\\gamma}(\\Omega_0)+t\\lambda_{p,\\gamma}(\\Omega_1)$; a violation for any $C^2$ convex bodies refutes Theorem 1.2. For Theorem 1.1, solve (1.1) on a $C^2$ convex domain with a flat boundary segment and inspect the Hessian of $w=-\\ln u$ at points where $\\nabla u\\neq0$; one negative eigenvalue of $D^2w$ refutes log-concavity.","supporting_citations":[{"cited_title":"Colesanti, P","cited_arxiv_id":null,"evidence_quote":"supplies the two comparison lemmas for sup-convolutions of strictly concave functions that Section 5 transfers to the inf-convolution setting."},{"cited_title":"Korevaar, Convex solutions to nonlinear elliptic and parabolic b oundary value problems, Indiana Univ","cited_arxiv_id":null,"evidence_quote":"supplies the concavity maximum principle applied to the regularized eigenfunctions to obtain convexity of $\\ln u_\\varepsilon$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the convex-analysis facts (inf-convolution regularity, subdifferential calculus, conjugates) used to build $\\tilde w$ and identify its critical set."},{"cited_title":"Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J","cited_arxiv_id":null,"evidence_quote":"provides the interior $C^{1,\\alpha}$ regularity estimates for quasilinear elliptic equations used for the eigenfunction and its regularizations."},{"cited_title":"Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm","cited_arxiv_id":null,"evidence_quote":"supplies the boundary regularity method via Schwarz reflection that gives global $C^{1,\\alpha}$ estimates."},{"cited_title":"Di Benedetto, C1+α local regularity of weak solutions of degenerate elliptic equations, N onlinear Anal","cited_arxiv_id":null,"evidence_quote":"supplies the local $C^{1,\\alpha}$ regularity theory for degenerate elliptic equations used in the regularity section."},{"cited_title":"Gilbarg and N","cited_arxiv_id":null,"evidence_quote":"supplies Schauder estimates and the Hopf and strong maximum principles used for boundary behavior and uniqueness proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the variational existence and bootstrap regularity framework for weak solutions of the eigenvalue problem."}],"review_version":1}