# zbMATH — the first resource for mathematics

An optimization problem in heat conduction. (English) Zbl 0668.49022
Given a surface $$\partial \Omega$$ in $${\mathbb{R}}^ 3$$, with prescribed temperature distribution $$\phi$$, surround $$\partial \Omega$$ with a prescribed volume of insulating material so as to minimize the loss of heat in a stationary situation. This is a generalization of the work of H. Alt and L. Caffarelli [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)] to nonconstant $$\phi >0$$. Analytically the authors prove existence and regularity for the following problem: minimize $$I(v)=\int_{\Omega}\Delta v$$ for all $$v\in H^ 1(\Omega)$$, $$v\geq 0$$ in $$\Omega$$, $$v=\phi$$ on $$\partial \Omega$$, $$\Delta$$ $$v\geq 0$$ in $$\Omega$$ and $$| \{v>0\}| \leq \mu <| \Omega |$$ for given $$\mu >0$$.
Reviewer: G.Dziuk

##### MSC:
 49M99 Numerical methods in optimal control 35D05 Existence of generalized solutions of PDE (MSC2000) 35J60 Nonlinear elliptic equations 35D10 Regularity of generalized solutions of PDE (MSC2000)
##### Keywords:
free boundary problem; heat conduction; existence; regularity
Full Text:
##### References:
 [1] N. Aguilera - H. Alt - L. Caffarelli , The optimal conductor problem , to appear. · Zbl 0588.49005 [2] H. Alt - L. Caffarelli , Existence and regularity for a minimum problem with regularity , J. Reine Angew. Math , 325 ( 1981 ), pp. 105 - 144 . MR 618549 | Zbl 0449.35105 · Zbl 0449.35105 [3] H. Alt - L. Caffarelli - A. Friedman , Variational problems with two phases and their free boundaries Trans. Amer. Math. Soc. , 282 ( 1984 ), pp. 431 - 461 . MR 732100 | Zbl 0844.35137 · Zbl 0844.35137 [4] I.I. Deniuuk , On integral functionals with a variable domain of integration , Proc. Steklov Inst. of Math. , 118 ( 1972 ), English transl. Amer. Math. Soc. ( 1976 ). MR 405198 | Zbl 0325.35002 · Zbl 0325.35002 [5] P. Garabedian , Partial Differentail Equations , Wiley , 1969 . Zbl 0124.30501 · Zbl 0124.30501 [6] D. Jerison - C. Kenig , Boundary Behavior of Harmonic functions in Non-Tangentially accessible domains , Adv. in math. 46 ( 1982 ), pp. 80 - 147 . MR 676988 | Zbl 0514.31003 · Zbl 0514.31003 [7] D. Kinderlehrer - L. Nirenberg - J. Spruck , Regularity in elliptic free boundary problems , I. J. d’Analyse Math. , 34 ( 1978 ), pp. 86 - 119 . MR 531272 | Zbl 0402.35045 · Zbl 0402.35045 [8] C.B. Morrey , Multiple integrals in the calculus of variations . Springer-Verlang , 1966 . Zbl 0142.38701 · Zbl 0142.38701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.