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Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents

https://doi.org/10.1515/ans-2022-0055

De Silva, Daniela; Savin, Ovidiu (January 2023, Advanced Nonlinear Studies)

Abstract We investigate the rigidity of global minimizers u ≥ 0 u\ge 0 of the Alt-Phillips functional involving negative power potentials ∫ Ω ( ∣ ∇ u ∣ 2 + u − γ χ { u > 0 } ) d x , γ ∈ ( 0 , 2 ) , \mathop{\int }\limits_{\Omega }(| \nabla u{| }^{2}+{u}^{-\gamma }{\chi }_{\left\{u\gt 0\right\}}){\rm{d}}x,\hspace{1.0em}\gamma \in \left(0,2), when the exponent γ \gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in R n {{\mathbb{R}}}^{n} are one-dimensional if γ \gamma is close to 2 and n ≤ 7 n\le 7 , or if γ \gamma is close to 0 and n ≤ 4 n\le 4 .
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On the Parabolic Boundary Harnack Principle

https://doi.org/10.1007/s44007-021-00001-y

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https://doi.org/10.1137/19M1308128

De Silva, Daniela; Savin, Ovidiu (January 2021, SIAM Journal on Mathematical Analysis)

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