Sharp regularity estimates for quasi-linear elliptic dead core problems and applications
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...[34, 35, 36, 39] for a similar strategy in dead core settings)....
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...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 > 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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"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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...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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"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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