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Journal ArticleDOI

Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

30 Apr 2018-Calculus of Variations and Partial Differential Equations (Springer Berlin Heidelberg)-Vol. 57, Iss: 3, pp 83
TL;DR: In this article, the authors studied geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type with strong absorption condition, and established a Liouville-type result for entire solutions provided that their growth at infinity can be controlled in an appropriate manner.
Abstract: In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type ( $$1< p< \infty $$ ) with strong absorption condition: $$\begin{aligned} -\mathrm {div}(\Phi (x, u, abla u)) + \lambda _0(x) u_{+}^q(x) = 0 \quad \hbox {in} \quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ where $$\Phi : \Omega \times \mathbb {R}_{+} \times \mathbb {R}^N \rightarrow \mathbb {R}^N$$ is a vector field with an appropriate p-structure, $$\lambda _0$$ is a non-negative and bounded function and $$0\le q0\} \cap \Omega $$ , where the regularity exponent is given explicitly by $$\gamma = \frac{p}{p-1-q} \gg 1$$ . Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of $$(N-1)$$ -Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the p-Laplace operator $$-\Delta _p u + \lambda _0 u^q\chi _{\{u>0\}} = 0$$ for any $$\lambda _0>0$$ .
Citations
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TL;DR: In this article, Da Silva, Joao Vitor, et al. present a paper on the evolution of Matematica in the context of the Consejo Nacional de Investigaciones Cientificas and Tecnicas.
Abstract: Fil: Da Silva, Joao Vitor. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

21 citations


Cites background from "Sharp regularity estimates for quas..."

  • ...[34, 35, 36, 39] for a similar strategy in dead core settings)....

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Journal ArticleDOI
TL;DR: In this article, the existence and simplicity of the first eigenvalue λ p and its corresponding eigenvector (u_p,v_p) for the following local/nonlocal PDE system were proved.
Abstract: In this work, given $p\in (1,\infty)$, we prove the existence and simplicity of the first eigenvalue $\lambda_p$ and its corresponding eigenvector $(u_p,v_p)$, for the following local/nonlocal PDE system \begin{equation}\label{Eq0} \left\{ \begin{array}{rclcl} -\Delta_p u + (-\Delta)^r_p u & = & \frac{2\alpha}{\alpha+\beta}\lambda |u|^{\alpha-2}|v|^{\beta}u & \mbox{in} & \Omega \\ -\Delta_p v + (-\Delta)^s_p v& = & \frac{2\beta}{\alpha+\beta}\lambda |u|^{\alpha}|v|^{\beta-2}v & \mbox{in} & \Omega u& =& 0&\text{ on } & \mathbb{R}^N \setminus \Omega v& =& 0&\text{ on } & \mathbb{R}^N \setminus \Omega, \end{array} \right. \end{equation} where $\Omega$$\subset$ $\mathbb{R}^N$ is a bounded open domain, $0

17 citations

Journal ArticleDOI
TL;DR: In this paper, the authors established sharp geometric regularity estimates for bounded solutions of a class of fully nonlinear elliptic equations with nonhomogeneous degeneracy, whose model equation is given by.
Abstract: We establish sharp $$C_{\text {loc}}^{1, \beta }$$ geometric regularity estimates for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy, whose model equation is given by $$\begin{aligned} \left[ |Du|^p+\mathfrak {a}(x)|Du|^q\right] {\mathcal {M}}_{\lambda , \Lambda }^{+}(D^2 u)= f(x, u) \quad \text {in} \quad \Omega , \end{aligned}$$ for a bounded and open set $$\Omega \subset {\mathbb {R}}^N$$ , and appropriate data $$p, q \in (0, \infty )$$ , $$\mathfrak {a}$$ and f. Such regularity estimates simplify and generalize, to some extent, earlier ones via totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear models in the theory of elliptic PDEs, which may have their own interest. We also deliver explicit examples where our results are sharp.

16 citations


Cites background or methods or result from "Sharp regularity estimates for quas..."

  • ...nd applied mathematics, such as in the study of blow-up analysis, related weak geometric and free boundary problems and for proving some Liouville type results, see [2], [12], [26], [27], [28], [30], [31], [33], [34], [56] and [57] for some enlightening examples. 1.1. Statement of the main results. In this section we will present some definitions, as well as some useful auxiliary results for our approa...

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  • ...g C1,α regularity estimates just along the a priori unknown set of singular points of solutions S0(u,Ω), where the “ellipticity of the operator” degenerates (see, for example, [26], [27], [28], [30], [31], [55], [56] and [57], where improved regularity estimates were addressed along certain sets of degenerate points of existing solutions). Finally, they are striking even for the simplest toy model: u ...

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  • ...utions to (4.1) along their touching ground boundary ∂{u &gt; 0} in contrast with Theorem 1.1. This is an important piece of information in several free boundary problems (cf. [26], [27], [28], [30], [31] and [57] for more explanations) Now, let us comment on the existence of a viscosity solution of the Dirichlet problem (4.1). Such an existence result follows by an application of Perron’s method sinc...

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Journal ArticleDOI
TL;DR: In this article, a quasi-linear elliptic operator with p-structure and concave-convex nonlinearities is considered, and the authors study existence and global uniform and explicit boundedness results to weak solutions.
Abstract: In this manuscript, we deal with an equation involving a combination of quasi-linear elliptic operators of local and non-local nature with p-structure, and concave–convex nonlinearities. The prototypical model is given by $$\begin{aligned} \left\{ \begin{array}{rclcl} -\Delta _p u + (-\Delta )^s_p u &{} = &{} \lambda _p u^q(x) + u^r(x) &{} \text{ in } &{} \Omega , \\ u(x)&{}>&{}0&{}\text{ in }&{} \Omega ,\\ u(x)&{} =&{} 0&{}\text { on } &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n$$ is a bounded and smooth domain, $$s\in (0,1)$$ , $$2 \le p < \infty $$ , $$0

15 citations

References
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Book
01 Jan 1992
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Abstract: GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems Differentiation of Radon Measures Lebesgue Points Approximate continuity Riesz Representation Theorem Weak Convergence and Compactness for Radon Measures HAUSDORFF MEASURE Definitions and Elementary Properties Hausdorff Dimension Isodiametric Inequality Densities Hausdorff Measure and Elementary Properties of Functions AREA AND COAREA FORMULAS Lipschitz Functions, Rademacher's Theorem Linear Maps and Jacobians The Area Formula The Coarea Formula SOBOLEV FUNCTIONS Definitions And Elementary Properties Approximation Traces Extensions Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Sobolev Functions Differentiability on Lines BV FUNCTIONS AND SETS OF FINITE PERIMETER Definitions and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions Isoperimetric Inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties of BV Functions Essential Variation on Lines A Criterion for Finite Perimeter DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS Lp Differentiability ae Approximate Differentiability Differentiability AE for W1,P (P > N) Convex Functions Second Derivatives ae for convex functions Whitney's Extension Theorem Approximation by C1 Functions NOTATION REFERENCES

5,769 citations


"Sharp regularity estimates for quas..." refers background in this paper

  • ...4) for balls contained in 3The reduced free boundary ∂red{u > 0} is subset of ∂{u > 0} where there exists the normal vector in the measure theoretic sense, see [18] for a survey about geometric measure theory....

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Journal ArticleDOI
TL;DR: On considere des solutions u∈H 1,p (Ω)∧L ∞ ( Ω) (1

1,463 citations


"Sharp regularity estimates for quas..." refers methods in this paper

  • ...[7], [17] and [34]) we get using the Divergence Theorem that ∫...

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Journal ArticleDOI
TL;DR: In this paper, it was shown that the Strong Maximum Principle is true for weak solutions of − Δu + β(u) = f with β a non-negative superharmonic continuous function in a domain Ω ⊂ ℝ� n�,n ⁽ 1,n ↽ 1.
Abstract: In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ω ⊂ ℝ n ,n ⩾ 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of − Δu + β(u) = f withβ a nondecreasing function ℝ → ℝ,β(0)=0, andf⩾0 a.e. in Ω if and only if the integral∫(β(s)s) −1/2 ds diverges ats=0+. We extend the result to more general equations, in particular to − Δ p u + β(u) =f where Δ p (u) = div(|Du| p-2 Du), 1

1,137 citations


"Sharp regularity estimates for quas..." refers methods in this paper

  • ...4) we obtain { div(Φ0(x0,∇u0)) = 0 in Ω u0(x0) = 0, and the Strong Maximum Principle from [35] implies that u0 ≡ 0....

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Journal ArticleDOI
TL;DR: In this article, the local C(1 + Alpha) nature of weak solutions of elliptic equations of the type (1.1) in the introduction under the degeneracy (or singularity) assumptions (A sub 1)-(A sub 3).
Abstract: : It is demonstrated the local C(1 + Alpha) nature of weak solutions of elliptic equations of the type (1.1) in the introduction under the degeneracy (or singularity) assumptions (A sub 1)-(A sub 3).

1,106 citations


"Sharp regularity estimates for quas..." refers methods in this paper

  • ...Finally, by invoking the uniform gradient estimates from [7], [17], [23, Chapter 4] and [34, Proposition 2] for bounded solutions we obtain that 1 r 1+q p−1−q |∇u(z)| = |∇ω(0)| ≤ C0....

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  • ...[7], [17] and [34]) we get using the Divergence Theorem that ∫...

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