On optimal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e41" altimg="si11.svg"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math> estimates for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e55" altimg="si12.svg"><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>-Laplace

Ding, Mengyao; Zhang, Chao; Zhou, Shulin

doi:10.1016/j.na.2020.112030

Cited by 10 publications

(4 citation statements)

References 25 publications

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“…For instance, we must mention the works of Kefi-Radulescu [9], Saoudi [10], Kefi-Saoudi [11], Xie-Chen [12], Nhan-Chuong-Truong [13], Zhang [14], Zhang-Motreanu [15], Yucedag [16], Yin-Li-Ke [17], Ait Hammou-Azroul-Lahmi [18], Ayazoglu-Ekincioglu [19], Ge-Wang [20], Ge-Lv [21], Ge-Radulescu [22], Heidarkhani-Afrouzi-Moradi-Caristi-Ge [23], Ge-Zhou-Wu [24], Fan [25] and the references therein. In relation to the regularity for solutions of differential equations with p(x)-Laplacian operator, we refer the readers to to [26][27][28][29][30][31][32][33][34][35][36], respectively, and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

Wang

Hou

2020

Mathematics

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In this paper, we consider the existence and uniqueness of solutions for a quasilinear elliptic equation with a variable exponent and a reaction term depending on the gradient. Based on the surjectivity result for pseudomonotone operators, we prove the existence of at least one weak solution of such a problem. Furthermore, we obtain the uniqueness of the solution for the above problem under some considerations. Our results generalize and improve the existing results.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

Wang

Hou

2020

Mathematics

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“…One difficulty is that the leading term in the Euler-Lagrange equation for curvature is singular (p < 2) or degenerate (p > 2) like the p-Laplacian (cf. [3,4,14,21]); in fact, the equation is formally given by where ∂ s denotes the arclength derivative. Thus k may not be of class C 2 and equation (1.1) may not be understood in the classical sense.…”

Section: Introductionmentioning

confidence: 99%

Complete classification of planar p-elasticae

Miura¹,

Yoshizawa²

2022

Preprint

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Euler's elastica is defined by a critical point of the total squared curvature under the fixed length constraint, and its L p -counterpart is called p-elastica. In this paper we completely classify all p-elasticae in the plane and obtain their explicit formulae as well as optimal regularity. To this end we introduce new types of p-elliptic functions which streamline the whole argument and result. As an application we also classify all closed planar p-elasticae. Contents 1. Introduction 1 2. The Euler-Lagrange equations 9 3. p-Elliptic functions 11 4. Classification, representation, and regularity of p-elasticae 25 5. Closed p-elasticae 39 Appendix A. First variation of the p-bending energy 45 References 47

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“…The regularity properties for minimizers of (1.4) or more general local variational functionals have been established in several works. See for instance [46,47,2,53,5,20,54,21,34,3,38,37,17,48,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Local regularity for nonlocal equations with variable exponents

Chaker¹,

Kim²

2021

Preprint

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In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler-Lagrange equations. We show that weak solutions are locally bounded when the variable exponent p is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on p. On the one hand, the class of admissible exponents is assumed to satisfy a log-Höldertype condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on p outside the domain.

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On optimal C1,α estimates for p(x)-Laplace

Cited by 10 publications

References 25 publications

Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

Complete classification of planar p-elasticae

Local regularity for nonlocal equations with variable exponents

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