A little help:

Suppose that the temperature T (x, y) of the sheet is limited. This condition is natural  if we take that observd  plate is boundary  case plate: 

                     --oo < x < +oo ,  0 < y < yo ,

whose upper edge, when yo increases, maitains a fixed temperature.                         

This problem can be mathematically formulated as follows:

Solve the Laplace’s equation

                           Txx(x.y)  + Tyy(x.y)  = 0   (--oo < x < +oo,  0 < y < +oo)          (1)

with boundary conditions  (2) :

                      T(x.0)  =  0,   for  I x I > 1  and    

                      T(x.0)  =  1,    for  I x I < 1 and                                                                                                                                                                                                IT(x.y) I 

This problem is concerned with solving Laplace’s differential equation applying conformal mappings, in our case it is

                             w =  ln (z-1) (z+1)-1                               (*)

(R.V. Churchill, Introduction to Complex Variables and Applications, Mc Grave-Hill Book Company, INC, 1948, pp. 209)  that the upper half-plane of (z) - plane mapps on the tape of width  p  of (w)-plane, ie. on the set of points

                    { (u,v) :  -oo < u < +oo,   0 < v < pi },   where

1)   the part of stright line  y = 0,  -1 < x < +1,  of (z) – plane, on which the potential       T = 1, passes in a straight line   v = pi,  -oo < u < +oo  in the (w)- plane, while

2)   two rays   y = 0,  x > 1   and   y = 0,  x < - 1  of (z) – plane respectively passes        in the positive and negative axis of the real axis of (w) – plane.

So, using well-known invariance of Laplace's differential equation in regard to conformal mappings, and appllying the conformal mapping (*), observed Dirichlet's problem (1) +(2) becomes the following Dirichlet's problem

                          Fuu + Fvv = 0,      -oo < u < +oo,   0 < v < pi                   (3)

with boundary conditions:

                           F(u,0)  =  0    and    F(u. pi)  =  1.                                   (4)

The solution of the  boundary problem (3) + (4)  is

                            F(u,v)  =   pi -1 v.

To obtain the solution of the starting boundary problem, it is necessary express v  through x and y, and so obtained  v  incorporated into the expression (5).  

Thanks that you find that this problem is interesting. I think that it is not only interesting but that  this method may be very useful for all who work in various fields of applied mathematics, especially in engineering mathematics and techniques.

Some of the references.that I highly recommended are

1. S. Farlow: Partial Differential Equations for Scientists and Engeneers, John Wiley and Sons, INC 1982.                                                                                              

2. A. R. 2. Forsayth: Theory of Partial Differential Equations,T. IV, Cambridge University Press, 1948.                                                                                            

3. F. John: Partial Differential Equations, Springer-Verlag, New-York Inc.1982, 4 th Ed

4. A.V. Bitsadze and D.F. Kalinichnko:.A Collection of Problems on the Equations of Mathematical Physics, Mir Publishers Moscow, 1980.

5, M. Vuković, Partial Diferentijalne Equations, Equations of mathematical Physics University Books, Sarajevo, 2001

Dear colleague Kultchitsky,

Problem of the solving the Dirichlet's problem for the upper semi-plane is very interesting and usefull. I asked the same question, and generally gave the answer, missing only some details at the end, (but if it is a problem they can be added)..

Using well-known invariance of Laplace differential equation in regard to conformal mappings, we exceed the Laplace's equation                                                                 Txx(x.y) + Tyy(x.y) = 0                                                                   to the Laplace's equations                                                                                                                Fuu + Fvv = 0,        

 can be found on the web-site

How to solve the Dirichlet's problem for the upper semi-plane ? - ResearchGate. Available from: https://www.researchgate.net/post/How_to_solve_the_Dirichlets_problem_for_the_upper_semi-plane [accessed May 18, 2016].

All best

Mirjana

wit

 (1) +(2) becomes the following Dirichlet's  problem

https://www.researchgate.net/post/How_to_solve_the_Dirichlets_problem_for_the_upper_semi-plane

Dear Mirjana,

This is an interesting topic.

Recall   the related  problem:

Solve the Laplace’s equation

                           Txx(x.y)  + Tyy(x.y)  = 0   (--oo < x < +oo,  0 < y < +oo)          (1)

with boundary conditions  (2) :

                      T(x.0)  =  0,   for  I x I > 1  and    

                      T(x.0)  =  1,    for  I x I < 1 and                                                                                                                                                                                        (2)        IT(x.y) I 

Dear Mirjana,

This is an interesting topic.

Consider the related problem: Solve the Laplace’s equation

Txx(x.y) + Tyy(x.y) = 0 (--oo < x < +oo, 0 < y < +oo) (1)

with boundary conditions (2) :

T(x.0) = 0, for I x I > 1 and

T(x.0) = 1, for I x I < 1 and |T(x.y) | < M, for --oo < x < +oo, where M is a positive constant.

This problem is known as the Dirichlet's problem for the upper semi-plane. It seems that we guess solution

$T(x,y)= pi-1  arg(z-1) (z+1)-1$. Is it correct?

Dear Miodrag,

Yes, of course it is  the Dirichlet's problem for the upper semi-plane. 

As you, I think that too, that it is very interesting topic, especially because we here use a conformal mappings and so we show not only the beauty, but also the power and importance of the complex analysis 

Excellent, you guess solution. I  didn't write until the end. I left the reader to finish or ask me a question. But, you are right, Namely, since                       

             w = u + iv = ln (z-1)(z+1)-1 = ln I(z-1)(z+1)-1I + i arg (z-1) (z+1)-1.  

when we take into account that

             arg (x + iy)  = arctg ( yx-1)

we obtain

             v =  arg (z - 1) (z+1)-1  = arg [ (x - 1) + iy ) ] [ (x + 1) + iy ) ]-1 =

                = arctg [ 2y ( x2 + y2 - 1 )-1 ].

Now, when we put this    v = v (x, y)    in    F(u,v) = pi -1 v,   we will get that the solution of our problem, known as the Dirichlet's problem for the upper semi-plane is:

              T( x, y)  = pi-1 arg(z-1) (z+1)-1          or

              T( x, y) = pi-1 arctg [ 2y ( x2 + y2 - 1 )-1]    (written in x, y coordinate ) 

Best regards and wishes, dear colleague Mateljevic

In connection with posed problem,   we   give the solution of Dirichlet's problem in the half-space. 

Suppose $f$ is continuous and bounded on $\mathbb{R}^{n-1}$.

Define $u$ on $\overline{H}$, where $\overline{H}$ is the half-space, by

$$ u(z) = \left\{\begin{array}{ll}

P_H [f](z) & \mbox{if}\,\, z\in H \\

f(z) & \mbox{if}\,\, z\in \mathbb{R}^{n-1}.

\end{array} \right.

$$

Then $u$ is continuous on $\overline{H}$ and harmonic on $H$.

Moreover, $|u|\leq |f|_\infty$ on $\overline{H}$.\\

See 7.3 in

S. Axler, P. Bourdon and W. Ramey: {\it Harmonic function theory},

Springer-Verlag, New York 1992.

Many thanks for your modern solution, although I, as someone who has except the study of mathematics completed three years of physics, prefer classic methods and think that are more beautiful, which come from intertwining of mathematics and physics.

But, for this there is another reason: my eminent professors and teachers: as assistant and follower of academicians Mahmut Bajraktarevic (doctorate  degree 1953, Sorbonne, Paris) , Fikret Vajzovic (pupil of Svetozar Kurepa from Zagreb and A. G. Kostychenko (in russian  Анатолий  Гордеевич Костюченко), and academician Manojlo Maravić one of the two first assistant at the Institute of SANU in Belgrade and Jovan Karamata’s assistant I belong to one of the best schools of the good old European analysis.

Dear Mirjana,

I also prefer classical methods and agree with you that ''are more beautiful, which come from intertwining of mathematics and physics''.

Roughly speaking, the book S. Axler, P. Bourdon and W. Ramey: {\it Harmonic function theory},

Springer-Verlag, New York 1992, is also related to classical methods in some extent.

It seems to me that we have interesting situation if we do not suppose that the solution of your original problem is bounded.

Here, I will give a few definition concerning my previous answer.

It is convenient to identify $\mathbb{R}^{n}$ with

$\mathbb{R}^{n-1}\times \mathbb{R}$ and (with analogy to

$2$-dimensional situation) to use notation $z=(x,y)$, where $x\in

\mathbb{R}^{n-1}$ and $y\in \mathbb{R}$.

We define $P_\mathbb{H}$ as function on $ \mathbb{H}\times \mathbb{R}^{n-1}$.

For $z=(x,y)$ and $t\in \mathbb{R}^{n-1}$

$$P_\mathbb{H} (z,t)= c_n \frac{y}{|z-t|^n},$$

where is $ c_n=2/(n V(B_n))$.

\begin{equation}\label{}

\int_{\mathbb{R}^{n-1}}P_\mathbb{H }(z,t) \, dt=1,\quad

z\in \mathbb{H }.

\end{equation}

For a complex Borel measure $\mu$ on $\mathbb{R}^{n-1}$, the

Poisson integral of $\mu$, denoted by $P_\mathbb{H} [\mu]$, is

\begin{equation}\label{}

P_\mathbb{H} [\mu](z): = \int_{\mathbb{R}^{n-1}}P_\mathbb{H} (z,t) \,d \mu(t)\, ,\quad

z\in \mathbb{H} .

\end{equation}

best,

Miodrag

If you agree with me, send me, please,  your solution in PDF form. I'm not in my at my apartment  (my mother is sick). I go to my haus only when I need something. My computer, books, all, ... is not with me for a long time.

I remembered academician M. Bajraktarević “from the forest of symbols”  something is not possible handle to manage. I think of your TEX.

But I have my books too: the good old - classical, from which I think younger scientists very often prescribed. (I question me whether an assistant professor, who has just begun to teach can write a book ?  But he/she must if want to go to a higher grade). 

I think that this books are:

1. S. Farlow: Partial Differential Equations for Scientists and Engineers, John Wiley& Sons Inc. 1982.

2. D,V, Churchill: Introduction to Complex Variables and Applications, McGrow - Hill Book Company, Inc., New York - Toronto - London, in 1948.

Dear Mirjana,

The corresponding   file is enclosed. I will make further corrections.

best,

Miodrag

Dear Miodrag,

I think it would be better that you  look my  books (I think S. Farlow and  D.V. Churchill) or to postpone the discussion when we see in Belgrade or in Sarajevo. Everything was there said clearly, brief, elegant, even with physical explanations.

I have dealt with this issue a long time ago, now only sometimes. My main concern now is algebra, and I currently organize one conference in Dubrovnik (International University Center - IUC 2016), more precisely:

The International Scientific Conference  „Graded Structures in Algebra and their applications” that will be dedicated to the memory of the famous French mathematician Marc Krasner, who himself made a huge contribution to the development of this theory, as well as abstract algebra and number theory in general. My Professor Krasner was the holder of the title of „Officier des Palmes de l'Académie des Sciences de Paris“ and the first laureate of the award „Doistau-Blutet de l'Académie des Sciences“ in the field of mathematics (for 1958). 

It is not easy when someone has a lot of different interests, but it is nice and interesting.

Best regards and wishes,

I hope, see you soon, Mirjana