Minimal Surface Equation
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Minimal surface equation mathematisation of a soap lm n+1 The minimal surface prob Consider a smo oth surface in IR representing the graph of a lem is also known as the function x ux x dened on a b ounded op en set in n+1 1 n n classical Plateau problem IR Assuming that u is suciently smo oth the area of the surface after the Belgian physicist is given by the nonlinear functional Joseph Antoine Ferdinand Z 12 Plateau who 2 Au jruj dx dx 1 n exp erimented by dipping wire contours into solutions 2 where ru is the gradient vector u x u x and jruj 1 n of soapy water and glycer ru ru ine Plateau did not have the mathematical skills to The minimal surface problem is the problem of minimising Au investigate the problem of sub ject to a prescrib ed b oundary condition u g on the b oundary existence of a minimal sur of To do this we consider the set U of all suciently g face theoretically Subse smo oth functions dened on that are equal to g on A quently Weierstrass Rie classical result from the calculus of variations asserts that if u is mann and Schwarz worked a minimiser of Au in U then it satises the EulerLagrange g on this question but it was equation not until the early s that the problem was nally 2 12 r ru jruj solved in a much more gener al form by Douglas and Rado In general there may b e one sev This quasilinear elliptic PDE is known as the minimal surface eral or no minimal surfaces spanning equation a given closed curve in space The theo Fig catenoid rem of Douglas and Rado asserts that as long In the case n can b e written as n+1 as is a closed Jordan curve in IR which is the 2 2 ju j u u u u ju j u b oundary of at least one surface with Au there exists a solution x y y x y xy y xx Trivially the plane u Ax B y C is a solution to this equa The study of minimal surfaces is an exciting and active eld of research which exploits a broad sp ec tion Nontrivial solutions including the helicoid and the catenoid trum of mathematical to ols from the theory of dierential equations Lie groups and Lie algebras dened by homologies and cohomologies b ordisms and varifolds as well as a range of computer visualisation and animation techniques Physically these surfaces have relevance to a variety of problems in q 1 1 2 2 materials science and civil engineering Of course they are most familiar as soap lmsbut not u tan y x u cosh a x y a soap bubbles for a bubble encloses air at a higher pressure than atmospheric and this changes to the capillary surface equation ref with a nonzero righthand side resp ectively where a is a constant were discovered over years ago Figs and In fact the catenoid arose in Eulers trea tise Methodus inveniendi lineas curvas maximi minimive propri etate graduentes which laid the foundations of the calculus References of variations The helicoid is the only nontrivial solution to U Dierkes S Hildebrandt A Kuster and O Wohlrab Minimal surfaces vols SpringerVerlag that is a harmonic function and the catenoid is the only nontrivial J Douglas Solution of the problem of Plateau Trans American Math So c 33 pp solution that is a surface of revolution The Scherk surface C C Grosjean and T M Rassias Joseph Plateau and his works in T M Rassias ed The problem of Plateau cos ay World Scientic u log a cos ax J B M C Meusnier Memoir sur la courbe des surfaces Memoirs de Savants Etrangers 10 J C C Nitsche Introduction to minimal surfaces Cambridge University Press discovered in where a is again a constant is the only non trivial solution to that can b e written in the form ux y R Osserman A survey of minimal surfaces Van Nostrand Reinhold f x g y T Rado On the problem of Plateau SpringerVerlag Fig helicoid c February Endre S uli .