2019/10/09 by Tommaso Di Marco, Di Marco, Tommaso, Paolo Marcellini +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1910.04158
We obtain an a-priori Wloc1,\∞ ( \Ω ;\ℝm )\n-bound for solutions in \Ω \⊂ \ℝn , n\≥ 2, to the\nelliptic system \
sumi=1n
frac
partial
partial\nxi
(
fracgt
( x,
|Du
|
)
|Du
| u_xi
alpha
)\n=0,
;
;
;
;
;
alpha =1,2,
ldots ,m, where g ( x,t ) ,\ng:\Ω \× [ 0,\∞ ) \→ [ 0,\∞ ) , is a\nCarath 'eodory function, convex and increasing with respect to the gradient\nvariable t\∈ [ 0,\∞ ) . We allow x-dependence, which turns out to\nbe a relevant difference with respect to the autonomous case and not only a\ntechnical perturbation. Our assumptions allow us to consider both fast and slow\ngrowth. We allow fast growth even of exponential type; and slow growth, for\ninstance of Orlicz-type with energy-integrands such as g ( x, | Du | )\n=|Du|\log (1+|Du|) or, when n=2,3, even asymptotic linear growth with energy\nintegrands of the type \g
( x,
| Du
|
) =
| Du
| -a
(\nx
)
sqrt
| Du
|
,. n