More Like this

1. The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

   https://doi.org/10.1007/s00526-021-01938-2  

   Banerjee, Agnid; Danielli, Donatella; Garofalo, Nicola; Petrosyan, Arshak (April 2021, Calculus of Variations and Partial Differential Equations)

   Abstract We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight$$|y|^a$$ ${\left|y\right|}^a$for$$a \in (-1,1)$$ $a\in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$(\partial _t - \Delta _x)^s$$ ${({\partial }_t-{\Delta }_x)}^s$for$$s \in (0,1)$$ $s\in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ $a=0$). 

   more » « less
2. On the Moduli of Lipschitz Homology Classes

   https://doi.org/10.1007/s12220-025-01916-6  

   Kangasniemi, Ilmari; Prywes, Eden (February 2025, The Journal of Geometric Analysis)

   Abstract We define a type of modulus$$\operatorname {dMod}_p$$ ${dMod}_p$for Lipschitz surfaces based on$$L^p$$ $L^p$-integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents$$p, q \in (1, \infty )$$ $p,q\in (1,\infty )$, every relative Lipschitzk-homology classchas a unique dual Lipschitz$$(n-k)$$ $(n-k)$-homology class$$c'$$ $c^{\prime }$such that$$\operatorname {dMod}_p^{1/p}(c) \operatorname {dMod}_q^{1/q}(c') = 1$$ ${dMod}_p^{1/p}\left(c\right){dMod}_q^{1/q}\left(c^{\prime }\right)=1$and the Poincaré dual ofcmaps$$c'$$ $c^{\prime }$to 1. As$$\operatorname {dMod}_p$$ ${dMod}_p$is larger than the classical surface modulus$$\operatorname {Mod}_p$$ ${Mod}_p$, we immediately recover a more general version of the estimate$$\operatorname {Mod}_p^{1/p}(c) \operatorname {Mod}_q^{1/q}(c') \le 1$$ ${Mod}_p^{1/p}\left(c\right){Mod}_q^{1/q}\left(c^{\prime }\right)\le 1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitzk-chains. 

   more » « less
3. Supersingular elliptic curves over ℤ _𝑝 -extensions

   https://doi.org/10.1515/crelle-2023-0029  

   Çiperiani, Mirela (June 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let $\mathrm{E}/\mathbb{Q}$\mathrm{E}/\mathbb{Q}be an elliptic curve and 𝑝 a prime of supersingular reduction for $\mathrm{E}$\mathrm{E}.Consider a quadratic extension $L/{\mathbb{Q}}_p$L/\mathbb{Q}_{p}and the corresponding anticyclotomic ${\mathbb{Z}}_p$\mathbb{Z}_{p}-extension $L_{\mathrm{\infty }}/L$L_{\infty}/L.We analyze the structure of the points $\mathrm{E}\left(L_{\mathrm{\infty }}\right)$\mathrm{E}(L_{\infty})and describe two global implications of our results. 

   more » « less
4. Construction of a Dirichlet form on Metric Measure Spaces of Controlled Geometry

   https://doi.org/10.1007/s11118-024-10144-6  

   Butaev, Almaz; Luo, Liangbing; Shanmugalingam, Nageswari (June 2024, Potential Analysis)

   Abstract Given a compact doubling metric measure spaceXthat supports a 2-Poincaré inequality, we construct a Dirichlet form on$$N^{1,2}(X)$$ $N^{1,2}\left(X\right)$that is comparable to the upper gradient energy form on$$N^{1,2}(X)$$ $N^{1,2}\left(X\right)$. Our approach is based on the approximation ofXby a family of graphs that is doubling and supports a 2-Poincaré inequality (see [20]). We construct a bilinear form on$$N^{1,2}(X)$$ $N^{1,2}\left(X\right)$using the Dirichlet form on the graph. We show that the$$\Gamma $$ $\Gamma$-limit$$\mathcal {E}$$ $E$of this family of bilinear forms (by taking a subsequence) exists and that$$\mathcal {E}$$ $E$is a Dirichlet form onX. Properties of$$\mathcal {E}$$ $E$are established. Moreover, we prove that$$\mathcal {E}$$ $E$has the property of matching boundary values on a domain$$\Omega \subseteq X$$ $\Omega \subseteq X$. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form$$\mathcal {E}$$ $E$) on a domain inXwith a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects. 

   more » « less
5. Reduction of the electroweak correlation in the PDF updating by using the forward–backward asymmetry of Drell–Yan process

   https://doi.org/10.1140/epjc/s10052-022-10280-6  

   Yang, Siqi; Fu, Yao; Liu, Minghui; Zhang, Renyou; Hou, Tie-Jiun; Wang, Chen; Yin, Hang; Han, Liang; Yuan, C. -P. (April 2022, The European Physical Journal C)

   Abstract We propose a new observable for the measurement of the forward–backward asymmetry$$(A_{FB})$$ $\left(A_{\mathrm{FB}}\right)$in Drell–Yan lepton production. At hadron colliders, the$$A_{FB}$$ $A_{\mathrm{FB}}$distribution is sensitive to both the electroweak (EW) fundamental parameter$$\sin ^{2} \theta _{W}$$ ${sin}^2{\theta }_W$, the weak mixing angle, and the parton distribution functions (PDFs). Hence, the determination of$$\sin ^{2} \theta _{W}$$ ${sin}^2{\theta }_W$and the updating of PDFs by directly using the same$$A_{FB}$$ $A_{\mathrm{FB}}$spectrum are strongly correlated. This correlation would introduce large bias or uncertainty into both precise measurements of EW and PDF sectors. In this article, we show that the sensitivity of$$A_{FB}$$ $A_{\mathrm{FB}}$on$$\sin ^{2} \theta _{W}$$ ${sin}^2{\theta }_W$is dominated by its average value around theZpole region, while the shape (or gradient) of the$$A_{FB}$$ $A_{\mathrm{FB}}$spectrum is insensitive to$$\sin ^{2} \theta _{W}$$ ${sin}^2{\theta }_W$and contains important information on the PDF modeling. Accordingly, a new observable related to the gradient of the spectrum is introduced, and demonstrated to be able to significantly reduce the potential bias on the determination of$$\sin ^{2} \theta _{W}$$ ${sin}^2{\theta }_W$when updating the PDFs using the same$$A_{FB}$$ $A_{\mathrm{FB}}$data. 

   more » « less