Question

Zeigen Sie, dass für eine zweimal stetig differenzierbare Lösung \(u(x, y)\) des Randwertproblems $$ -\Delta u=f \quad \text { in } \Omega, \quad u=0 \quad \operatorname{auf} \Gamma_{d}, \quad \frac{\partial u}{\partial \mathbf{n}}=q \quad \operatorname{auf} \Gamma_{n} $$ die Integralgleichung $$ \int_{\Omega}[\nabla u \cdot \nabla h-f h] d F-\int_{\Gamma_{n}} q h d s=0 $$ gilt, wobei \(f, q\) gegebene, integrierbare Funktionen sind und \(\Gamma=\Gamma_{d} \cup \Gamma_{n}, \Gamma_{d} \cap\) \(\Gamma_{n}=\emptyset\), der Rand von \(\Omega\) ist. \(h\) sei stetig differenzierbar und auf \(\Gamma_{d}\) gleich Null. \(\frac{\partial}{\partial \mathrm{n}}\) bezeichnet die Ableitung in Richtung der äußeren Normalen auf dem Rand \(\operatorname{von} \Omega\).

Short answer

The integral equation holds due to the weak form derived from the PDE and boundary conditions.

Step-by-step solution

Understand the Problem Statement

We are given a boundary value problem involving the Laplacian \(-\Delta u = f\) in a region \(\Omega\) with boundary conditions: \(u = 0\) on \(\Gamma_d\), and \(\frac{\partial u}{\partial \mathbf{n}} = q\) on \(\Gamma_n\). We need to show that the given integral equation holds for a twice continuously differentiable solution \(u(x,y)\).

Set Up the Weak Form of the Problem

To derive the weak form, multiply the PDE by a test function \(h\), integrate over \(\Omega\), and apply integration by parts. The problem's weak form is: \[ \int_{\Omega} (-\Delta u) h \, dF = \int_{\Omega} f h \, dF. \]

Apply Integration by Parts

Using integration by parts, which states \(\int_{\Omega} \Delta u h \, dF = -\int_{\Omega} abla u \cdot abla h \, dF + \int_{\Gamma} \frac{\partial u}{\partial \mathbf{n}} h \, ds\), leads to: \[ -\int_{\Omega} abla u \cdot abla h \, dF + \int_{\Gamma} \frac{\partial u}{\partial \mathbf{n}} h \, ds = \int_{\Omega} f h \, dF. \]

Substitute Boundary Conditions

Substitute the boundary conditions into the integrated form. Use \(u = 0\) on \(\Gamma_d\) which implies \(h = 0\) on \(\Gamma_d\), and \(\frac{\partial u}{\partial \mathbf{n}} = q\) on \(\Gamma_n\) to get the expression: \[ -\int_{\Omega} abla u \cdot abla h \, dF + \int_{\Gamma_n} q h \, ds = \int_{\Omega} f h \, dF. \]

Conclusion of the Integral Equation

Rearrange the integrated form as initially stated: \[ \int_{\Omega} [abla u \cdot abla h - f h] \, dF - \int_{\Gamma_n} q h \, ds = 0. \] This shows that the integral equation holds given the boundary conditions and properties of \(u\) and \(h\).

Key concepts

Laplacian Operator

The Laplacian operator is a crucial component in mathematical physics and analysis. It is denoted as \( \Delta \) and often appears in boundary value problems, like the one we are tackling here. The Laplacian is a second-order differential operator given by the divergence of the gradient of a function. In two dimensions, for a function \( u(x, y) \), it is expressed as: \[\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}.\]

• The Laplacian operator measures how much the function \( u \) deviates from its mean value at a particular point.
• It is widely used in physical problems where diffusion, wave propagation, or potential fields are involved.
• In our boundary value problem, the condition \( -\Delta u = f \) signifies an equilibrium state, with \( f \) acting as a source term.

Understanding the Laplacian operator is vital, as it relates directly to the physical phenomena being modeled. Here, solving \( -\Delta u = f \) helps us deduce how \( u \) behaves under the specified conditions across the domain \( \Omega \).

Weak Formulation

Weak formulation is a modern mathematical approach to solving differential equations, particularly boundary value problems. Instead of focusing solely on the pointwise satisfaction of the differential equation, weak formulation emphasizes an average or integrated form of the equation over a domain.This concept involves introducing a test function, denoted \( h \), which belongs to a set of appropriately smooth functions. We multiply the original differential equation by \( h \) and integrate over the domain \( \Omega \). This transforms the problem into:\[\int_{\Omega} (-\Delta u)h \, dF = \int_{\Omega} f h \, dF.\]

• This technique helps in deriving solutions that are smooth within \( \Omega \) but need not be differentiable everywhere.
• It also accommodates more complex boundary conditions and irregular domains.
• The test function \( h \) vanishes on the boundary \( \Gamma_d \), simplifying some terms in the process.

The weak formulation is particularly beneficial for numerical approximations, such as finite element methods, where exact solutions may be cumbersome or difficult to obtain analytically.

Integration by Parts

Integration by parts is an essential technique in calculus, especially when dealing with boundary value problems and converting differential equations into their weak forms.It facilitates the transformation of differential operators into boundary terms and simplified integrals. The integration by parts formula in the context of a boundary value problem is:\[\int_{\Omega} \Delta u h \, dF = -\int_{\Omega} abla u \cdot abla h \, dF + \int_{\Gamma} \frac{\partial u}{\partial \mathbf{n}} h \, ds.\]

• This method essentially 'transfers' a derivative from one function to another within an integral.
• When dealing with the Laplacian \( \Delta u \), integration by parts helps manage the terms involving gradients and divergence.
• The boundary term \( \int_{\Gamma} \frac{\partial u}{\partial \mathbf{n}} h \, ds \) becomes key, since it connects the behavior of \( u \) at the boundary with the external conditions.

By applying integration by parts in our problem, we effectively reduce complexity, reconstituting our differential equation into an integral form which is easier to work with, especially when using test functions that fulfill specific boundary conditions.