Which boundary conditions lead to existence and uniqueness of a second order PDE by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16www.cns.gatech.edu/~predrag/courses/PHYS-6124-12/StGoChap6.pdf 6.1 "Classification of PDE's" clarifies which boundary conditions are needed for existence and uniqueness of each type of second order of PDE:
This idea comes up particularly in the phase space coordinate of Hamiltonian mechanics.
Specifies fixed values.
Can be used for elliptic partial differential equations and parabolic partial differential equations.
Numerical examples:
Specifies the derivative in a direction normal to the boundary.
Can be used for elliptic partial differential equations and parabolic partial differential equations.
Sets both a Dirichlet boundary condition and a Neumann boundary condition for a single part of the boundary.
Can be used for hyperbolic partial differential equations.
We understand intuitively that this imposes stricter requirements on solutions, which makes it easier to guarantee uniqueness, but also harder to have existence. TODO intuitively why hyperbolic need this extra level of restriction.
Linear combination of a Dirichlet boundary condition and Neumann boundary condition at each point of the boundary.
Examples:
- In this case, the normal derivative at the boundary is proportional to the difference between the temperature of the boundary and the fixed temperature of the external environment.The result as time tends to infinity is that the temperature of the plaque tends to that of the environment.
In the context of wave-like equations, an open-boundary condition is one that "lets the wave go through without reflection".
This condition is very useful when we want to simulate infinite domains with a numerical method. Ciro Santilli wants to do this all the time when trying to come up with demos for his physics writings.
Here are some resources that cover such boundary conditions:
- www.asc.tuwien.ac.at/~arnold/pdf/graz/graz.pdf lots of slides
- hplgit.github.io/wavebc/doc/pub/._wavebc_cyborg002.html mentions them and gives a 1D formula. It mentions that things get complicated in 2D and 3D TODO why.The other page: hplgit.github.io/wavebc/doc/pub/._wavebc_cyborg003.html shows solution demos.
Multiple boundary conditions for different parts of the boundary.
Most commonly, boundary conditions such as the Dirichlet boundary condition are taken to be fixed values in time.
This basically adds one more ingredient to partial differential equations: a function that we can select.
And then the question becomes: if this function has such and such limitation, can we make the solution of the differential equation have such and such property?
Control theory also takes into consideration possible discretization of the domain, which allows using numerical methods to solve partial differential equations, as well as digital, rather than analogue control methods.
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