A class of De Giorgi type and Hölder continuity

Fan, Xiaohui; Zhao, Dun

doi:10.1016/s0362-546x(97)00628-7

Cited by 241 publications

(183 citation statements)

References 15 publications

Supporting

Mentioning

181

Contrasting

Order By: Relevance

“…In the Dirichlet case, this can be deduced from Theorem 7.1 of Ladyzhenskaya-Uraltseva [28] (problems with standard growth conditions) and Theorem 4.1 of Fan-Zhao [17] (problems with nonstandard growth conditions). However, in the Neumann case, the aforementioned theorems cannot be used since they require that u| ∂ is bounded (u being the weak solution).…”

Section: Two Auxiliary Resultsmentioning

confidence: 99%

Anisotropic nonlinear Neumann problems

Gasiński

Papageorgiou

2011

Calc. Var.

View full text Add to dashboard Cite

We consider nonlinear Neumann problems driven by the p(z)-Laplacian differential operator and with a p-superlinear reaction which does not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). In the process, we also prove two results of independent interest. The first is about the L ∞ -boundedness of the weak solutions. The second relates W 1, p(z) and C 1 local minimizers. Mathematics Subject Classification (2000)

show abstract

Section: Two Auxiliary Resultsmentioning

confidence: 99%

Anisotropic nonlinear Neumann problems

Gasiński

Papageorgiou

2011

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…The careful scrutiny of the presentation in [28] reveals the dependance of c and κ on sup u and structure constants (cf. Lemma 3.5).…”

Section: Remark 32mentioning

confidence: 99%

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Adamowicz

Lundström

2015

Annali di Matematica

View full text Add to dashboard Cite

We investigate various boundary decay estimates for p(·)-harmonic functions. For domains in R n , n ≥ 2 satisfying the ball condition (C 1,1 -domains), we show the boundary Harnack inequality for p(·)-harmonic functions under the assumption that the variable exponent p is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson-type estimate for p(·)-harmonic functions in NTA domains in R n and provide lower and upper growth estimates and a doubling property for a p(·)-harmonic measure.

show abstract

“…In [1,2,13], this result was used to prove-under optimal conditions on the regularity of p(x)-(partial) Hölder continuity results for minimizers and also their gradient. Note that in [24], Hölder continuity for solutions of equations was shown by a generalization of DeGiorgi's methods, under optimal assumptions on the regularity of p (·). We remark at this point that in order to guarantee that the solution u of an equation of p(x) growth type is Hölder continuous, i.e., to show C 0,α -regularity, it is necessary to impose logarithmic Hölder continuity of the exponent…”

Section: P(·)(1+δ) Locmentioning

confidence: 99%

Gradient estimates via non standard potentials and continuity

Bögelein

Habermann

2010

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

Abstract. We consider elliptic problems with non standard growth conditions whose most prominent model example is the p(x)-Laplacean equationwith a measure data right-hand side µ. We prove pointwise gradient estimates in terms of a non standard version of the non-linear Wolff potential of the right-hand side measure, and moreover a characterization for C 1 -regularity of the solution, also in terms of the Wolff potential. The C 1 -regularity criterion is also related to the density of µ and the decay rate of its L n -norm on small balls. Moreover, from the pointwise gradient estimates the Calderón and Zygmund theory and several types of local estimates follow as a consequence.

show abstract

A class of De Giorgi type and Hölder continuity

Cited by 241 publications

References 15 publications

Anisotropic nonlinear Neumann problems

Anisotropic nonlinear Neumann problems

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Gradient estimates via non standard potentials and continuity

Contact Info

Product

Resources

About