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1. The "hot spots" conjecture on the Vicsek set

   https://doi.org/10.1515/dema-2019-0003  

   Ionescu, Marius; Savage, Thomas L. (February 2019, Demonstratio Mathematica)

   Abstract We prove the “hot spots” conjecture on the Vicsek set. Specifically, we will show that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary. 

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2. Inverse Iteration for the Laplace Eigenvalue Problem With Robin and Mixed Boundary Conditions

   https://doi.org/10.46787/pump.v8i.4261  

   Lyons, Benjamin; Ruttenberg, Ephraim; Zitzelberger, Nicholas (May 2025, PUMP journal of undergraduate research)

   We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal eigenfunction and show that the corresponding Rayleigh quotients converge to the principal eigenvalue. We also propose a related iterative method for an eigenvalue problem arising from a model for optimal insulation and provide some partial results. 

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3. Inverse Problem of Travel Time Difference Functions on a Compact Riemannian Manifold with Boundary

   https://doi.org/10.1007/s12220-018-00111-0  

   de Hoop, Maarten V.; Saksala, Teemu (January 2019, The Journal of Geometric Analysis)

   We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differential structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points. 

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4. Inverse Problem for Compact Finsler Manifolds with the Boundary Distance Map

   de Hoop, Maarten V; Ilmavirta, Joonas; Lassas, Matti; Saksala, Teemu. (January 2019, ArXiv.org)

   We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differential structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points. 

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5. The regular free boundary in the thin obstacle problem for degenerate parabolic equations

   Banerjee, A.; Danielli, D.; Garofalo, N.; Petrosyan, A. (June 2020, Algebra i analiz)

   In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial_t - \Delta_x)^s$$ for $$s \in (0,1)$$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $$H^{1+\gamma,\frac{1+\gamma}{2}}$$ regularity of the free boundary near such regular points. 

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