$ C^{1, \alpha} $-regularity theory for weak solutions of a general quasilinear elliptic equation

Fengping Yao; Fengping Yao

doi:10.3934/cam.2026018

Communications in Analysis and Mechanics

2026, Volume 18, Issue 2: 444-461. doi: 10.3934/cam.2026018

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Theory article

$ C^{1, \alpha} $-regularity theory for weak solutions of a general quasilinear elliptic equation

Fengping Yao ^,

Department of Mathematics, Shanghai University and Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, China

Received: 15 September 2025 Revised: 26 February 2026 Accepted: 18 March 2026 Published: 19 May 2026
35B65, 35J60

In this paper, we establish $ C^{1, \alpha} $-regularity theory with an accurate estimate
$ \begin{eqnarray*} \| u\|_{C^{1, \alpha}(B_{1/2})} \le C \left(b^{-1}\left( \|f\|_{L^\infty(B_1)}\right)+ \|u\|_{W^{1, B}(B_1)}\right), \end{eqnarray*} $
where $ b(t) = ta(t) $, for weak solutions of the following general quasilinear elliptic equation with Orlicz growth in divergence form:
$ \begin{eqnarray*} -\text{div} \; \! \big( a(|\nabla u|)\nabla u \big) = f\in L_{loc}^\infty(\Omega) \quad \mbox{in} \; \; \Omega \subset \mathbb{R}^n \end{eqnarray*} $
for $ n \ge 2 $. Its prototypes are the nonhomogeneous elliptic $ p $-Laplacian equations with and without a logarithmic term, respectively. Meanwhile, we also present the local optimal $ (1+ s_a') $-cap continuity for the above problem.
- $ C^{1, \alpha} $,
- regularity,
- divergence,
- weak solutions,
- Orlicz,
- quasilinear,
- elliptic
Citation: Fengping Yao. $ C^{1, \alpha} $-regularity theory for weak solutions of a general quasilinear elliptic equation[J]. Communications in Analysis and Mechanics, 2026, 18(2): 444-461. doi: 10.3934/cam.2026018

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Abstract

In this paper, we establish $ C^{1, \alpha} $-regularity theory with an accurate estimate

$ \begin{eqnarray*} \| u\|_{C^{1, \alpha}(B_{1/2})} \le C \left(b^{-1}\left( \|f\|_{L^\infty(B_1)}\right)+ \|u\|_{W^{1, B}(B_1)}\right), \end{eqnarray*} $

where $ b(t) = ta(t) $, for weak solutions of the following general quasilinear elliptic equation with Orlicz growth in divergence form:

$ \begin{eqnarray*} -\text{div} \; \! \big( a(|\nabla u|)\nabla u \big) = f\in L_{loc}^\infty(\Omega) \quad \mbox{in} \; \; \Omega \subset \mathbb{R}^n \end{eqnarray*} $

for $ n \ge 2 $. Its prototypes are the nonhomogeneous elliptic $ p $-Laplacian equations with and without a logarithmic term, respectively. Meanwhile, we also present the local optimal $ (1+ s_a') $-cap continuity for the above problem.

References

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