I have a region defined like this:

    circle = Disk[{4.5, 3}, 0.5];
    pin = Rectangle[{4, 0}, {5, 3}];
    square = Rectangle[{0, 0}, {9, 9}];
    region = RegionDifference[square, RegionUnion[circle, pin]];

Applying RegionPlot[region] gives this:

[![Region][1]][1]

Now I need to setup boundary conditions for this region the following way:

1) Top, left, right walls: u[x,y] == 0

2) Bottom wall 0 <= x < 4: u[x,y] == 0

3) Bottom wall 5 < x <= 0: u[x,y] == 0

4) Wall at x = 4, for 0 <= y < 3: u[x,y] == 10

5) Wall at x = 5, for 0 <= y < 3: u[x,y] == 10

6) Semicircle with the center at x = 4.5 and y = 3 (radius = 0.5): u[x,y] == 10

These boundary conditions should be applied to a Laplace equation:

    sol = NDSolveValue[{D[u[x, y], x, x] + D[u[x, y], y, y] == 0,
    bc},
       u, {x, y} \[Element] region]
    
    DensityPlot[sol[x, y], {x, y} \[Element] region, Mesh -> None, 
     ColorFunction -> "Rainbow", PlotRange -> All, 
     PlotLegends -> Automatic]

_____

Update 1: the result should be something like this:

[![image2][2]][2]

That is what I received when I ran the code provided by user21 on Mathematica 10.3. I introduced:

`mesh = ToElementMesh[DiscretizeRegion[region], 
   MaxCellMeasure -> 0.01];`

and in plotting I changed `Mesh -> All` (for the picture on the left)

_____

Update 2: User21 provided a new part of the code:

    DensityPlot[sol[x, y], {x, -10, 10}, {y, -10, 10}, Mesh -> All, 
     ColorFunction -> "Rainbow", PlotRange -> All, 
     PlotLegends -> Automatic, MaxRecursion -> 4]

 It gives the same plot as what you can see in User21's answer but only if you use Mathematica of the version newer than 10.3. For the version 10.3 I get an error *"InterpolatingFunction::dmval: "Input value {-9.99857,-9.99857} lies outside the range of data in the interpolating function. Extrapolation will be used."* And the plot looks like this:

[![error][3]][3]

It gets a bit better if I switch `{x, -10, 10}, {y, -10, 10}` to `{x, y} \[Element] region` but still the plot looks unacceptable:

[![error2][4]][4]


  [1]: https://i.sstatic.net/19geQ.png
  [2]: https://i.sstatic.net/S6CYf.png
  [3]: https://i.sstatic.net/E8mNg.png
  [4]: https://i.sstatic.net/P1wqn.png