On a Variational Approach for Water Waves

Abstract

Using a variational method due to Alt and Caffarelli [1], we study regularity and qualitative properties of local and global minimizers of a functional with a variable domain of integration related to water waves.

On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

“…Symmetric minimizers and their free boundaries. The existence of symmetric minimizers in the sense of Definition 1.1 was previously observed in Theorem 5.10 in [9] (see also Theorem 1.6 and Remark 5.14 [23]). In particular, we recall that for a symmetric minimizer u ∈ K γ , the portion of the free boundary ∂{u > 0} in {−λ/2 < x < 0} can be described by the graph of a function x = g(y), where…”

“…Although many results on water waves have been obtained by mapping the domain of the fluid into a fixed domain in the complex plane by means of a hodograph transform (see, for example, [5,6,31,32,34] and the references therein), in recent years variational approaches have been proposed to tackle these kind of problems (see, for example, [9,23,35,36,39,40]). The advantages of considering a variational formulation of problem (1.9) are twofold: it allows for more general geometries such as multiple air components, while at the same time it retains the physical intuition of the model.…”

“…Proof. Since u 0 defined in (1.7) belongs to K γ and is such that J h (u 0 ) < ∞, the solvability of the minimization problem for J h in K γ follows from Theorem 1.3 in [2] (see also Theorem 2.2 in [9]). The proofs for statements (i) through (iv) can also be found in [2]; more precisely, we refer the reader to Lemma 2.…”

On the existence of non-flat profiles for a Bernoulli free boundary problem

“…The next theorem shows that the reduced free boundary of a global minimizer u is regular except for the case in which supp u ⊂ R × [0, h] and supp u ⊂ R × [0, h). Part (ii) significantly improves the understanding of the so-called non-physical solutions in [AL12].…”

“…(1.6) if a local minimizer u has support below the line {y = h} and if there exists a point x 0 = (x 0 , h) ∈ ∂{u > 0}, then |∇u(x)| ≤ Cr 1/2 , for x ∈ B r (x 0 ) (1.7) (see [AL12,Remark 3.5]). On the other hand, using a monotonicity formula and a blow up method, Varvaruca and Weiss in [VW11, Theorem A] proved that for a suitable definition of solution if the constant C in (1.7) is one then the rescaled function u(x 0 + rx) r 3/2 → √ 2 3 ρ 3/2 cos 3 2 min max θ, − 5π 6 , − π 6 + π 2 as r → 0 + , strongly in W 1,2 loc (R 2 ) and locally uniformly on R 2 , where (x, y) = (ρ cos θ, ρ sin θ), and near x 0 the free boundary ∂{u > 0} is the union of two C 1 graphs with right and left tangents at x 0 (see also [WZ12]).…”