Boundary Unique Continuation on $$C^1$$-Dini Domains and the Size of the Singular Set
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DOI: 10.1007/s00205-022-01771-7
Abstract: Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to… read more here.
Keywords: set boundary; unique continuation; boundary unique; dini ... See more keywords
Convex functionals and the stratification of the singular set of their stationary points
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DOI: 10.1016/j.aim.2018.08.008
Abstract: We prove partial regularity of stationary solutions and minimizers $u$ from a set $\Omega\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_\Omega F(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies… read more here.
Keywords: stratification; convex functionals; singular set; set stationary ... See more keywords
New ε-regularity criteria and improved box-counting dimension of potential singular set for suitable weak solutions to the Navier-Stokes-Planck-Nernst-Poisson system
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DOI: 10.1088/1402-4896/adf8a2
Abstract: Since the seminal work of (Caffarelli et al 1982 Comm. Pure Appl. Math.35 771—831) on the partial regularity theory for the three-dimensional incompressible Navier-Stokes equations, where they proved that the one-dimensional parabolic Hausdorff measure of… read more here.
Keywords: box counting; counting dimension; navier stokes; dimension ... See more keywords