Gradient regularity in mixed local and nonlocal problems

Abstract

Minimizers of functionals of the type $$\begin{aligned} w\mapsto \int _{\Omega }[|Dw|^{p}-fw]\,\textrm{d}x+\int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\frac{|w(x)-w(y)|^{\gamma }}{|x-y|^{n+s\gamma }}\,\textrm{d}x\,\textrm{d}y\end{aligned}$$ w ↦ ∫ Ω [ | D w | p - f w ] d x + ∫ R n ∫ R n | w ( x ) - w ( y ) | γ | x - y | n + s γ d x d y with $$p, \gamma>1>s >0$$ p , γ > 1 > s > 0 and $$p> s\gamma $$ p > s γ , are locally $$C^{1, \alpha }$$ C 1 , α -regular in $$\Omega $$ Ω and globally Hölder continuous.

Fine boundary regularity for fully nonlinear mixed local-nonlocal problems

“…Also see [49] for its parabolic version. For HJBI-type integro-PDEs, interior C 1,η -regularity is established by Mou and Zhang [44] and for mixed local nonlocal fractional p-Laplacian, see [22]. The C 1,η -regularity up to the boundary for linear mixed local-nonlocal operators is recently obtained in [11].…”

Regularity results for mixed local and nonlocal double phase functionals

“…One of main points in our problem is the interplay between local and nonlocal phenomena in the double phase structure, which gives rise to several difficulties in combining the local theory with the nonlocal theory. Indeed, the approaches in the present paper are different from those in [9,26]. On one hand, since the second term in (1.1) is of a local nature, we could not derive the same form of the logarithmic estimate as in [9,Lemma 5.1].…”

“…On one hand, since the second term in (1.1) is of a local nature, we could not derive the same form of the logarithmic estimate as in [9,Lemma 5.1]. On the other hand, since (1.1) features purely nonlocal behaviors on the set {a(x) = 0}, we could not compare (1.1) with a local problem as in [26]. We thus develop a different method motivated from the ones in [1,3,23], whose crucial tools include the expansion of positivity results described in Lemmas 4.2 and 4.3 below.…”

“…Subsequently, in [35] the purely analytic approaches in [29,30] were applied to mixed local and nonlocal p-Laplacian type equations. For more on regularity and other qualitative behaviors, see [5,6,10,26,36,47]. In particular, the maximal regularity in [26] was achieved for general problems modeled after…”

“…For more on regularity and other qualitative behaviors, see [5,6,10,26,36,47]. In particular, the maximal regularity in [26] was achieved for general problems modeled after…”