Local boundedness for solutions of a class of nonlinear elliptic systems

2021/08/26 by G. Cupini, Cupini, G., F. Leonetti +3
Computer Science · Mathematics · #35J47 (Primary) 35B65 (Secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2108.11813

openalex publication_date 2021/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of m equations in divergence form, satisfying p growth from below and q growth from above, with p ≤ q; this case is known as p, q-growth conditions. Well known counterexamples, even in the simpler case p=q, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component u^α of the solution u=(u¹,...,u^m) satisfies an improved Caccioppoli's inequality and we get the boundedness of u^α by applying De Giorgi's iteration method, provided the two exponents p and q are not too far apart. Let us remark that, in dimension n=3 and when p=q, our result works for (3)/(2) < p < 3, thus it complements the one of Bjorn whose technique allowed her to deal with p ≤ 2 only. In the final section, we provide applications of our result.

Related

• Local boundedness for weak solutions to some quasilinear elliptic systems
• Regularity and explicit L^∞ estimates for a class of nonlinear elliptic systems
• Boundary regularity for elliptic systems under a natural growth condition
• Regularity for nonuniformly elliptic equations with p,q-growth and explicit x,u-dependence
• Partial regularity result of elliptic systems with Dini continuous coefficients and q-growth