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We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and partial regularity of volume-constrained minimizers. We also derive the Euler--Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal $frac12$-derivative of the capacitary potential.
We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator with several local conservation laws (local density, momentum and energy). We exhibit all equilibria and entropy-maximizing special modes, a
In this paper we prove existence, uniqueness and regularity of certain perturbed (subsonic--supersonic) transonic potential flows in a two-dimensional Riemannian manifold with convergent-divergent metric, which is an approximate model of the de Laval
The Benjamin Ono equation with a slowly varying potential is $$ text{(pBO)} qquad u_t + (Hu_x-Vu + tfrac12 u^2)_x=0 $$ with $V(x)=W(hx)$, $0< h ll 1$, and $Win C_c^infty(mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $$Q_{
We consider several limiting cases of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent $gamma$ (called the $gamma$-ensembles). The effective potential, which is essentially
For an upstream supersonic flow past a straight-sided cone in $R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the