Optimal uniform bounds for competing variational elliptic systems with variable coefficients

2023/02/16 by Manuel Dias, Hugo Tavares, Dias, Manuel +1
Computer Science · Mathematics · #35B09 #35B45 #35B65 #35J47 #35J91 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2302.08254

openalex publication_date 2023/02/16 · openalex created_date 2023/02/19 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ ℝ^N be an open set. In this work we consider solutions of the following gradient elliptic system -div(A(x)∇ u_i,β) = f_i(x,u_i,β) + a(x)β|u_{i, β}|^γ-1u_{i, β} \mathop∑_j=1^l_{j≠ i} |u_{j, β}|^{γ+ 1}, for i=1,…, l. We work in the competitive case, namely β<0. Under suitable assumptions on A, a, f_i and on the exponent γ, we prove that uniform L^∞-bounds on families of positive solutions \u_β\_β<0=\(u_1,β,…, u_l,β)\_β<0 imply uniform Lipschitz bounds (which are optimal). One of the main points in the proof are suitable generalizations of Almgren's and Alt-Caffarelli-Friedman's monotonicity formulas for solutions of such systems. Our work generalizes previous results, where the case A(x)=Id (i.e. the operator is the Laplacian) was treated.

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