Summary
The weak form of the free boundary problem for an axisymmetric partially penetrating well may be formulated as follows: find ϕ(r) ε C0([r0, r1]) and\(u \in C^0 (\bar \Omega ) \cap \mathop {V^1 }\limits^ - (\Omega )\) such that
and u satisfies appropriate boundary conditions. Here, u is related to the hydraulic head, ϕ(r) is the unknown water-air interface, Ω is the region of saturated flow
K1 is a convex set in the weighted Sobolev space V1(Ω).
We reduce the problem to three families of variational inequalities by using a type of « Baiocchi transform », study equivalence of the three families and regularity of the solutions of the variational inequalities. Finally, we prove the exictence of the solution for the well problem.
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Sponsored by the National Science Foundation under Grant No. MCS77-26732 with support fac lities provided by the U.S. Army under Contract No. DAAG29-80-C-0041.
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Cryer, C.W., Zhou, S.Z. The solution of the free boundary problem for an axisymmetric partially penetrating well. Annali di Matematica pura ed applicata 135, 219–235 (1983). https://doi.org/10.1007/BF01781069
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DOI: https://doi.org/10.1007/BF01781069