Regularity results for nonlocal equations = 비국소 방정식의 정칙성 결과

국문 초록 (Abstract)

이 학위논문에서는 비선형 비국소 방정식의 다양한 정칙성 결과를 조사
한다. 먼저 비선형 비국소 방정식에 대한 수많은 해의 측정을 확립한다. 다
음으로는 적절한 분수 형태로 표현되는 우변에 의존하는 해의 극대 소볼레프
정칙성을 증명한다.

다국어 초록 (Multilingual Abstract)

We investigate several regularity results of nonlinear nonlocal equations. First,
we establish numerous interrelated gradient estimates in the nonlinear nonlocal setting.
Second, we prove the optimal higher Sobolev regularity of the solution depending on a regularity assumption of the right-hand side presented by a suitable fractional operator.

목차 (Table of Contents)

• Abstract 1
• 1 Introduction 1
• 1.1 Nonlinear nonlocal equations with a quadratic growth and nondivergence data 2
• 1.2 Nonlinear nonlocal equations with a p-growth and fractional divergence type data 4
• 2 Preliminaries 7
• 2.0.1 Some elementary estimates 9
• 2.0.2 Some embedding results 9
• 2.0.3 Some properties of difference quotients 11
• 2.0.4 A covering lemma 18
• 2.0.5 Fractional De Giorgi classes 19
• 3 Gradient estimates for nonlinear nonlocal equations with a quadratic growth 22
• 3.1 Gradient H¨older regularity for homogeneous equations 25
• 3.2 Oscillation decay and higher differentiability of the gradient 43
• 3.3 Potential estimates at the gradient level 61
• 4 Optimal fractional Sobolev regularity of solutions to s-fractional p-Laplacian equations 70
• 4.1 Comparison estimates 76
• 4.2 Coverings of upper level sets 84
• 4.3 Lq-estimate of dsu 103
• 4.3.1 Restricted range of fractional differentiability 104
• 4.3.2 Improved range of fractional differentiability 118
• Abstract (in Korean) 142
• Acknowledgement (in Korean) 143