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1.2 Boundary Potentials and Calder´on’s Projector 3

c
Here v p denotes a particular solution of (1.1.1) in Ω or Ω and f has been
n
c
extended from Ω (or Ω ) to the entire R . Moreover, for the extended f
c
we assume that the integral defined in (1.1.5) exists for all x ∈ Ω (or Ω ).
Clearly, with this particular solution, the boundary conditions for u are to
be modified accordingly.
Now, without loss of generality, we restrict our considerations to the so-
lution u of the Laplacian (1.1.6) which now can be represented in the form:
∂u ∂E(x, y)

u(x)= E(x, y) (y) − u(y) ds y . (1.1.7)
∂n ∂n y
y∈Γ y∈Γ
For given boundary data u| Γ and ∂u |
∂n Γ , the representation formula (1.1.7)
defines the solution of (1.1.6) everywhere in Ω. Therefore, the pair of bound-
ary functions belonging to a solution u of (1.1.6) is called the Cauchy data,
namely

u| Γ
Cauchy data of u := . (1.1.8)
∂u
|
∂n Γ
In solid mechanics, the representation formula (1.1.7) can also be derived
by the principle of virtual work in terms of the so–called weighted resid-
ual formulation. The Laplacian (1.1.6) corresponds to the equation of the
equilibrium state of the membrane without external body forces and vertical
displacement u. Then, for fixed x ∈ Ω, the terms

∂E(x, y)
u(x)+ u(y) ds y
∂n y
y∈Γ
correspond to the virtual work of the point force at x and of the resulting
boundary forces ∂E(x, y)/∂n y against the displacement field u, which are
equal to the virtual work of the resulting boundary forces ∂u |
∂n Γ acting against
the displacement E(x, y), i.e.

∂u
E(x, y) (y)ds y .
∂n
y∈Γ
This equality is known as Betti’s principle (see e.g. Ciarlet [42], Fichera [75]
and Hartmann [121, p. 159]). Corresponding formulas can also be obtained
for more general elliptic partial differential equations than (1.1.6), as will be
discussed in Chapter 2.

1.2 Boundary Potentials and Calder´on’s Projector

The representation formula (1.1.7) contains two boundary potentials, the
simple layer potential