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Mathematics LibreTexts

1.3: Partial Differential Equations

( \newcommand{\kernel}{\mathrm{null}\,}\)

The same procedure as above applied to the following multiple integral leads to a second-order quasilinear partial differential equation. Set

$$E(v)=\int_\Omega\ F(x,v,\nabla v)\ dx,\]

where ΩRn is a domain, x=(x1,,xn), v=v(x): ΩR1, and v=(vx1,,vxn). Assume that the function F is sufficiently regular in its arguments. For a given function h, defined on Ω, set

$$V=\{v\in C^2(\overline{\Omega}):\ v=h\ \mbox{on}\ \partial\Omega\}.\]

Euler equation. Let uV be a solution of (P), then

ni=1xiFuxiFu=0

in Ω.

Proof. Exercise. Hint: Extend the above fundamental lemma of the calculus of variations to the case of multiple integrals. The interval (x0δ,x0+δ) in the definition of ϕ must be replaced by a ball with center at x0 and radius δ.

Example 1.2.2.1: Dirichlet integral

In two dimensions the Dirichlet integral is given by

$$D(v)=\int_\Omega\ \left(v_x^2+v_y^2\right)\ dxdy\]

and the associated Euler equation is the Laplace equation u=0 in Ω.

Thus, there is natural relationship between the boundary value problem

$$\triangle u=0\ \ \mbox{in}\ \Omega,\ u=h\ \ \mbox{on}\ \ \partial\Omega\]

and the variational problem

$$\min_{v\in V}\ D(v).\]

But these problems are not equivalent in general. It can happen that the boundary value problem has a solution but the variational problem has no solution, for an example see Courant and Hilbert [4], Vol. 1, p. 155, where h is a continuous function and the associated solution u of the boundary value problem has no finite Dirichlet integral.

The problems are equivalent, provided the given boundary value function h is in the class
H1/2(Ω), see Lions and Magenes [14].

Example 1.2.2.2: Minimal surface equation

The non-parametric minimal surface problem in two dimensions is to find a minimizer u=u(x1,x2) of the problem

$$\min_{v\in V}\int_\Omega\ \sqrt{1+v_{x_1}^2+v_{x_2}^2}\ dx,\]

where for a given function h defined on the boundary of the domain Ω

$$V=\{v\in C^1(\overline{\Omega}):\ v=h\ \mbox{on}\ \partial\Omega\}.\]

Comparison surface

Figure 1.2.2.1: Comparison surface

Suppose that the minimizer satisfies the regularity assumption uC2(Ω), then
u is a solution of the minimal surface equation (Euler equation) in Ω
x1(ux11+|u|2)+x2(ux21+|u|2)=0.

In fact, the additional assumption uC2(Ω) is superfluous since it follows from regularity considerations for quasilinear elliptic equations of second order, see for example Gilbarg and Trudinger [9].

Let Ω=R2. Each linear function is a solution of the minimal surface equation (???). It was shown by Bernstein [2] that there are no other solutions of the minimal surface quation. This is true also for higher dimensions n7, see Simons [19].

If n8, then there exists also other solutions which define cones, see Bombieri, De Giorgi and Giusti [3].

The linearized minimal surface equation over u0 is the Laplace equation u=0. In R2 linear functions are solutions but also many other functions in contrast to the minimal surface equation. This striking difference is caused by the strong nonlinearity of the minimal surface equation.

More general minimal surfaces are described by using parametric representations. An example is shown in Figure 1.2.2.21. See [18], pp. 62, for example, for rotationally symmetric minimal surfaces.

Rotationally symmetric minimal surface
Figure 1.2.2.2: Rotationally symmetric minimal surface

1An experiment from Beutelspacher's Mathetikum, Wissenschaftsjahr 2008, Leipzig

Neumann type boundary value problems

Set V=C1(¯Ω) and

$$E(v)=\int_\Omega\ F(x,v,\nabla v)\ dx-\int_{\partial\Omega}\ g(x,v)\ ds,\]

where F and g are given sufficiently regular functions and ΩRn is a bounded and sufficiently regular domain.
Assume u is a minimizer of E(v) in V, that is

$$u\in V:\ \ E(u)\le E(v)\ \ \mbox{for all}\ v\in V,\]

then

Ω (ni=1Fuxi(x,u,u)ϕxi+Fu(x,u,u)ϕ) dxΩ gu(x,u)ϕ ds=0


for all ϕC1(¯Ω). Assume additionally uC2(Ω), then u is a solution of the Neumann type boundary value problem
ni=1xiFuxiFu=0  in Ωni=1Fuxiνigu=0  on Ω,

where ν=(ν1,,νn) is the exterior unit normal at the boundary Ω. This follows after integration by parts from the basic lemma of the calculus of variations.

Example 1.2.2.3: Laplace equation

Set

$$E(v)=\frac{1}{2}\int_\Omega\ |\nabla v|^2\ dx-\int_{\partial\Omega}\ h(x)v\ ds,\]

then the associated boundary value problem is

u=0  in Ωuν=h  on Ω.

Example 1.2.2.4: Capillary equation

Let ΩR2 and set

$$E(v)=\int_\Omega\ \sqrt{1+|\nabla v|^2}\ dx+\frac{\kappa}{2}\int_\Omega\ v^2\ dx -\cos\gamma\int_{\partial\Omega}\ v\ ds.\]

Here κ is a positive constant (capillarity constant) and γ is the (constant) boundary contact angle, i. e., the angle between the container wall and the capillary surface, defined by v=v(x1,x2), at the boundary.

Then the related boundary value problem is

div (Tu)=κu  in ΩνTu=cosγ on Ω,


where we use the abbreviation

$$Tu=\frac{\nabla u}{\sqrt{1+|\nabla u|^2}},\]

div (Tu) is the left hand side of the minimal surface equation (???) and it is twice the mean curvature of the surface defined by z=u(x1,x2), see an exercise.

The above problem describes the ascent of a liquid, water for example, in a vertical cylinder with cross section Ω. Assume the gravity is directed downwards in the direction of the negative x3-axis. Figure 1.2.2.3 shows that liquid can rise along a vertical wedge which is a consequence of the strong non-linearity of the underlying equations, see Finn [7]. This photo was taken from [15].

alt
Figure 1.2.2.3: Ascent of liquid in a wedge

Contributors and Attributions


This page titled 1.3: Partial Differential Equations is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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